He's Got Eyes for the Job
We don't really get to know a whole lot about Patrick, aside from the fact that he doesn't seem to have much luck with the ladies. He's convinced that Mary hates him, but Mary seems so sweet that we don't see how this could be the case… until Mary comes over and is so blatantly unamused by his googly-eyed glasses that we realize he's definitely right.
When Patrick finally does get a girlfriend, he's way excited about it. The only trouble is that his new girlfriend is a former patient—yeah, it's totally Clementine.
Basically, Patrick spent one single night watching over Clementine's unconscious body, and he fell in love with her. And then he stole her panties. A real winner, this guy.
In the end, of course, nothing actually works out for Patrick. He takes his mimicking of Joel too far and gets burned. Honestly, it's kind of hard to feel sorry for the guy. He may be just another lonely dude desperate for a woman in his life, but he's also a total creep. Maybe he's just never heard of Tinder?
Also, did you notice what happens to Patrick's eyes when Joel sees him in his dream? They are freaky and distorted in some way that's hard to see; they may be inverted. That's a bad sign right there, showing us that something is pretty wrong with this guy.
In the end, Patrick doesn't necessarily change the course of the film or how we think about it; he's just another incentive for Joel to keep his memories. He also highlights some of the possible ethical problems inherent in memory erasure and mind invasion.