As far as I know, canon never made it clear how exactly Voldemort realized he was connected to Harry. We see how Harry sees visions of what Voldemort is doing and feeling, but does Voldemort see and feel Harry's in return?

Unlike Harry, Voldemort doesn't have a piece of someone else's soul in him, and he seemed ignorant of his Horcruxes' destruction, so it seems unlikely he'd be any more aware of the piece of soul in Harry.

Was he simply detecting Harry's presence inside his mind? If so, why did it take him over a year from the first vision in GoF to notice despite his skill at legilimency and occlumency?

 

Relevant quotes:

and

Plus, Dumbledore:

also

Basically: Yes, Voldemort can peer into Harry's mind. No, he doesn't do it, because it is immensely painful for him, as he figured out when he attempted to possess Harry. He views their connection as mostly a disadvantage, which is is - Harry can gain valuable intel from him, but the reverse isn't true - so he uses Occlumency to prevent sharing from happening. He wasn't aware of the extent of their connection until halfway through the fifth year, when Harry had the vision of the snake. Harry interrupts Snape's explanation, but it appears that this was a particularly jarring intrusion, and that's why he finally noticed.

 

And voldemort would likely think this as the mark of equal so he didn't have a reason to think that it is a piece of his soul in harry scar.

 

How much did Voldemort really knew about the connection though, besides the fact that he had access to Harry's mind? Did he wonder about the source, the mechanics, or a way to destroy it? He used Occlumecy, yes, but we don't see him looking for information about it. He just seems to accept that there is a connection. Then again, cannon was mostly Harry-centric, so it is possible that he had some ideas...

 

Knowing how the connection came to be or how it worked wouldn't of helped Voldemort win the war. Going through the effort of destroying it seems pointless if he could just ignore it using occlumency. I think the phrase "out of sight out of mind" describes the connection perfectly.

 

I think that voldemort never knew the full extent of this connection to harry potter, that is why he never got ride of it or investigated. If he knew the full extent he would have investigated it further. His personality would never allow someone to have that kind of connection to him and access to this inner most thoughts.

 

He definitely investigated it. His duel with Harry in the graveyard made him realize that there was more to the prophecy than just Harry's "destiny" to vanquish him. The connection between their minds solidified that curiosity. His desire for the prophecy was him investigating that connection.