does the internal fan in the laptop still work? If that has failed, it'd explain the heat issues.

 

Many a laptop I have seen suffer the heat death. From what I know its usually a combination of battery/power supply degradation and the fan getting old and weak. I don't have technical knowledge however, so that's mostly just guesswork and observation. My last laptop seemed to have problems with both, and eventually multiple parts fried into uselessness (the CD drive, battery, and ethernet were all nonfunctional by the end).

 

Things could be clogged up with dust? Take a blower and clean out of the path through with your lappy disperses heat. I usually notice a significant decline in heat issues every time I do this.

Also, your graphics card might have some integrated fan or heat sink, or something. Check that, too.

 

This is just a guess, but my thinking was as the battery degrades it gets to be more and more of a heat source, and does a lot of damage by frying the hell out of everything even as it sits there as a useless lump. Does it feel really hot while the laptop is operating (even though its basically useless)? For that old computer, I just took it out and relied on plug power 100%, though I don't know how much good it did.

 

From my personal experience:
Check if the laptop's fan is not clogged up with dust etc. I had exactly the same problem, after two years mine had a "carpet" 1/10 inch thick beneath the fan. Remove the case/panel if possible, unscrew the fan and clean it thoroughly. Replace everything as it was and it's done.
If that does not help, there is a possibility that the heatsink doesn't work as it should - the thermal grease could have lost it's properties (it requires replacement every 2 to 5 years). It's a bit more complicated and more prone to fatal screw-ups. Just remove the heatsink, clean the remains of thermal grease carefully (if it's silver-based alcohol + cotton sticks works just fine). After that apply a new layer of thermal grease (just a drop and use a credit card or sth to spread it into a thin layer) and put back everything into it's proper place.

 

Bukay's got the right idea (also TheFineBalance)- here's a link for a support forum that includes a link to the service manual. There's no way a well-made laptop heats up that much in the first year without interference from, say, cat hair.

Forum link: http://forum.notebookreview.com/asus-reviews-owners-lounges/456659-asus-n61jq-owners-lounge-397.html

Service guide link: http://www.gentechpcforums.com/Images/Asus/manuals/N61VN Series Chapter 02-v1.0.pdf

Also- buy a new battery.

 

This. When your battery degrades it becomes a solid block of heat as it is 'continually' in charge. Just like all rechargeable devices, when in charge, there is some heat created from whatever reaction is needed to recharge it. I had my old laptop die from this as the HDD cracked from the heat.

My current laptop has also had the battery die. It was like yours where the charge wouldn't hold and had a nice little red cross. Removing the battery is good and created far less heat for the whole system. I also invested in a cheap laptop fan base to augment my laptops current cooling system. The only drawback to this is sometimes because of how effective the fan base is, the in-built fan of the laptop may switch off as the temperature seems to be well handled without its use.

 

Well you could run you laptop with the battery removed, and see if that fixes it like every one has bee saying. But most likely your problem has to with dust clogging up the vents, or some sort of an internal heat sink bust. Just call Asus, if you're under warranty.