For most games I play, I usually won't go after every trophy or achievement. I will probably look at the list and look at certain ones that may be easy to do in an area that may be coming up, but I usually just take it in stride. Block an enemy's attack 50 times? So easy, I might as well go out and get it. Kill 10 people in a row using only stealth attacks? Meh, if it happens it happens.
But for the games that REALLY get me gripped, I feel compelled to get the trophies. Its usually all about the character settings in that world that I get so sucked into it. That's not to say that I usually get everything though. I absolutely love Nier, but I never got all the trophies even though I probably could have. I have most of the achievements for Mass Effect 3, and yet I've never played the game on Insanity. I was really into Heavy Rain though, and I managed to get everything for that game. I already know that if the Kingdom Hearts 1.5 ReMix comes over stateside, I'm totally picking it up despite having the games on the PS2 and whatnot? Why? The trophies. I love the series a lot, so I know I can probably get a lot of the trophies. Same thing if I got a PS Vita and I obtained Persona 4 Golden.
More often than not though, I get close to getting all the trophies and achievements because when I beat the game, I'm already so close. Lego games for instance are a big easy platinum, and for Marvel Ultimate Alliance 2 it was really simple to have most of the trophies already.
Still, I usually have a set of games where I say, "There are more Trophies/Achievements to get, but I feel very finished with the game". Using Nier as an example again, I COULD probably go back and get the last 3 or 4 trophies, but I felt I got JUST enough of the game from the trophies I already got. Another game, Transformers War for Cybertron has a bunch of Multiplayer trophies in its list. Aside from just a couple, I simply focused on getting all of the singleplayer trophies. To me, I feel like I actually got all the trophies in the game, even though I clearly didn't. I look at these lists, and I feel fulfilled despite not doing everything.
And then of course there are the games where just ask for a bit more of your time than you are willing to give, usually for no good reward besides being the one stopping you from getting all the trophies/achievements. Flag collecting in AC1 or some of the achievements in Gears of War 3 come to mind.
TL;DR: If its easy, really immersive, and I'm pretty much there, I'll try to get every trophy or achievement. However, sometimes I feel like I did enough in regards to trophies or achievements (in a good way), and I won't go for every one. It varies from game to game and depends on the achievement list. Super hard, time consuming, or just plain annoying ones I would more than likely not get.