The very definition of the genre so to speak, is for Harry to become strong and independent, something radically different than Harry's canon personality. So in your opinion how do you do it properly?

 

I wrote a sarcastic answer, then figured I should probably actually answer your question...

Spoiler

Go to the goblins, remember Griphook and tip him for good service. Be surprised when it turns out he's actually Hob of the Goblins, and swears a solemn blood vow to serve your family for the rest of eternity. On the way out, pop in for an inheritance ceremony, and find out that actually you're Lord Potter-Black-Weasley-Slytherin-Gryffindor-Peverall-Ambrosius-xxxShadowxxx (but just call me Harry, I hear Lord Potter and look for my father!)

Black(through Sirius)
Weasley(through a lost card game)
Slytherin(conquest)
Gryffindor(direct descendant)
Peverall(same)
Ambrosius(because you're the first person with a magical power of 9001 to get tested)
xxxShadowxxx.

Once leaving the bank, trip over Fleur, and fall on top of her, your penis slips inside. Her burst of allure makes everyone else on the street orgasm then and there except you,so she falls in love with you, and displays her feathers. Seeing you still love a feathery person, Tonks admits to never showing her true form, but that she can with you. Luna is also there. Padma and Parvati are purebloods back over 3000 years, but aren't white so don't count, and they also want your dick. Cho and Angelina join in because then you can show you're not racist. Hermione/Ginny/OC fills the last slot, although you might also take on extra names depending on whether women can carry names through generations, in which case expect Daphne, Tracey, more OCs, Gabrielle, and Sinistra to join in too. Also Pansy was hiding being attractive because otherwise Slytherins would make her the house Sluterin, and she was mean to everyone because Draco made her.

After swearing a mass marriage in the middle of Diagon Alley, you all go shopping using your newfound money. Don't forget an extended trunk you can all live in, and to add all the charms that go onto magical tents too. Then it's time for a new wand without tracking charms, which a handy vampire in Knockturn will whip up for you out of a Basilisk tooth and wizard blood. You might consider adding thestral hair too.

You'll then retire to your country estate, before popping out for a bit of hob-nobbing with Lizzie. She'll swear the UK Marine Corp (actually the corps of Royal Marines, but hey-ho), the Gurkha (and they'll take Harry out drinking), and an entire battalion of muggleborn trained by squibs, because they've had ten years to prepare since "old whatsisface" popped his clogs. She'll give you her personal blessing to carry guns, knight you, and make you the sworn wand of the Crown. Don't forget to wave the magna-carta around a load too.

Having swung by home for a few rounds of sex with all your new wives, you'll then drop into the Wizengamot, break a stalemate 200 years old, and re-enact reforms that change the face of magical Britain forever. They'll probably look a lot like modern US government, but that's because it's a magical kingdom of fun and happiness for everyone.

Then more sex. Tonks will look like Fleur for a bit, and Padvati and Parme will join in for some double-double action, and twincest is hot.

Next day, you go and leave your house, preparing to fight against Voldemort

Oh, and making all house elves free goes somewhere. Because the majority are more like Dobby than Winky.

Also, don't forget the country estate is in either Wales or Scotland, because England is shit, and everyone speaks Gaelic, because the author is Welsh (their greatx3 grandparent came from Queensferry).

AUTHOR'S NOTE
I'm sorry UwU, i cant think of what happens next. Vote in the poll to see who Harry adss to his Harem, or if he should fight Moldyshorts next chapter!!!!

The big thing a lot of people seem to forget when making an independent Harry is the reason he's going that way. Normally, it's chafing under restrictions from Dumbledore.

Putting himself in a position of subservience to the muggle/goblin government/crown doesn't gel with that. Especially because Harry doesn't actually have a plan for Voldemort. If he had a plan that would likely work, and instead everyone was ignoring him completely, then you could go that way. The Denarian series by Shezza I feel is a good starting place - Harry actually doesn't need help with his plans, and frequently obstructs Dumbledore's plans/gets his own knocked back. However, it doesn't feature Harry going off on his own completely.

But how do you make Harry strong?

He's a teenager. And a teenager without much in the way of knowledge - he's above average in his grades (compared to muggle exams at the time), but not genius level. He can't practice much in the way of magic over the summer (depending on how the trace works, but you can circumvent that). And most non-magical means of becoming stronger aren't much use - he could practice dodging,or running, or weightlifting. But AoE spells are a thing, and being able to do really fucking heavy deadlifts doesn't help much when someone's spitting curses at you.

That, I think, is where most indifics break down.

They make it seem like all it takes is 8-10 weeks of reading books and practising spells most days puts you on a level with (or, normally, better than) aurors and teachers. This is often done in self study, although it might be focused on "dark/illegal/family" spells, but with a tutor, suddenly Harry is the best wizard to ever walk the planet.

But the reality is that Voldemort has fifty years on Harry. Hell, let's say he remembers nothing of the ~15 years he spent as a ghost, and nothing of the ~20 years he spent not knowing magic, then learning it... And he still has 15 years of doing high level magic against Harry.
Harry hasn't actually graduated from high school by the time he tends to go "indy". It's the equivelent of a 15 year old sneaking out to smoke cigarettes and complain about how they hate their parents.

To make it believable, Harry needs a tutor. They need to be willing to teach him, but also to actually push him. They themselves need to actually be a powerful wizard. Somehow, this needs to avoid going through Harry being kept in the dark from them. This often brings a crossover.
Time turners would help, but they start the descent into the really daft tropes that plague the genre. Instead, maybe have Harry brew sleeping and wakefulness potions, so he gets by on 2 hours sleep, and increases his ability with potions.
Ideally, you would have Harry be as canon, but if you have to, maybe he could have done some research into obscure magics at Hogwarts - nothing that would show up in canon, but esotoric and mostly useless spells that he liked the sound of - if you were sent into a magical world, would you want to immerse yourself in it? Maybe he learned a teeth-cleaning spell, or a shoe tying charm, and so on. So he's got a wider base. This is often a place we get a crossover.
You could give Harry a different 'base', but it doesn't kick in until he's 15, for example, Harry the Kryptonian. But you then have to figure out all the extra parts of that, and whether you do more crossover, or just the single piece.

In theory, I'm fine reading something that ticks all the generic Indific cliches, as long as the writing is up to snuff.

Generally, it isn't.

 

It's easier to say how not to write Indy!Harry.

1. Independent is not the same thing as special. Independent Harry does not need unique abilities, equipment or titles.
2. Independent from what? From Dursleys, Hogwarts, the ministry, his friends?
3. Independence is not the same thing as doing everything alone.
4. "Training" is never interesting to read about. Neither is reading. Learning by doing is a lot better.

 

Look at the stories that had a good, strong Harry.

Skitter Leap, By the Divining Light/Canloadh's Song, Circular Reasoning, An Unfound Door, Heartlands/Wastelands of Time, Sitra Ahra, Harry Potter and the Boy Who Lived, Ectomancer, The Song of the Trees. I know these aren't necessarily "independent" Harry, but they are good and feature a strong protagonist. Independent Harry can be established by just not making him follow Dumbledore for whatever reason you wish, preferably not because Dumbledore is evil. Now compare what they have in common and how they established Harry in their stories.

 

This bit right here actually got me curious - and intrigued - to see a story where Orginal!Voldemort didn't actually survive the rebound Killing Curse and end up as a snake on the back of some dim wit's head. Instead, I think it'd be awesome to see a Dairy!Voldemort be the one to make the big come back and to become the primary antagonist of the fic. Mostly because he'd be young and without all the knowledge - and could potentially end up going a different way in terms of tactics and such.

Sorry for not adding anything to the discussion, OP. Just a random thought that struck my mind while reading @Thaumologist response.

Edit: Also, it'd put Harry and Voldemort on more even footing in terms of experience and knowledge. Someone would have to figure out how to outplay around Dumbledore, but it'd be a good opportunity to show the genius of Tom that was hinted at in canon but never really shown. He just powers through stuff in the books, in my opinion.

 

The best Indy Harry I can think of is featured in Renegade Cause--this fic isn't without problems, but it executed the indy thing really well, imo.

It begins with the inciting incident, which pushes Harry to do something that's in-character for canon Harry, but results in serious consequences, which pushes Harry to do something he wouldn't typically do. And it keeps going. Harry involves himself in adult schemes, so he has to up his game, rinse and repeat--and thus you get a series of events wherein Harry goes from canon to indy in a logical way. He becomes good--very good--but doesn't instantly get on Voldemort's level.

Indy Harry is, I think, less about "hurr durr, I need to become independent" and more about the character making decisions that keep escalating the stakes--and then exploring how Harry deals with the consequences of his choices.

 

If I were to write an Indy!Harry, I would just do it without all the extraneous bullshit authors seem to tag on. It's possible to start Harry on the path to independant freedom with just what we have in canon, so you don't need secret power blocks on Harry's magic, Dumbledore stealing from his bank account or him collecting an inheritance that just happens to solve all his problems. No friendly goblins, no custom wands or trunk, and certainly no elemental powers upon turning 17. All you need is Harry just realizing that he's likely going to die if he doesn't get some help, and actually wanting to do something about it.

 

I think the major problem I have with Indy!Harry fics - beyond just the general low level of writing that seems to be the norm - is in how they write Harry's relationships and interactions with any would-be authority figures, particularly Dumbledore. People seem to take independent in this context to just mean disrespectful and arrogant. Harry is no longer Dumbledore's man, so obviously he has to treat Dumbledore like total shit and act like a complete cunt whenever they're together. It's this weirdly childish view of independence that's more about narcissism and insular group think than anything else. Harry, and whoever else is on his 'side', are the genius renegades, and anyone else is totally incompetent at best or outright villainous. Real independence doesn't really look anything like that, you still rely on other people and respect them, especially when they're obviously qualified and talented in ways that you aren't, you just don't act completely subordinate to them. Most Indy!Harry authors don't even try to strike that balance, they just hurl themselves down the road of Super!Harry & Incompetent!EveryoneElse

 

Indy Harry means Harry shouldn't be under the thumb of an authority figure - in fics where Dumbledore is some kind of wise, godlike, benificent person, Harry is restricted - he will always be if not following Dumbledore's orders, respecting him and taking his advice. Any fic where Dumbledore is not taken down a peg at least to the extent that he has made a great many serious, unforgivable mistakes over the years (to the degree that he would be criminally negligent in a court) not only ignores Rowlings plot holes but also means that if you want the fic to be comprehensible, Harry has to listen to him almost to the degree of working for him. Which does not an 'independent' fic make.
When he is portrayed as selfish and manipulative or even evil it gives Harry a secondary antagonist beyond Voldemort and can make for a compelling story. I think it just adds the drama of a story to have such betrayals, though it can get hackneyed after a while. With a plausible reason for Harry no longer listening to Dumbledore in terms of the fight against Voldemort in particular, Harry becomes a truly independent protagonist and the powerful, self-reliant hero (or anti-hero) that we like to see.