I will generally just play a game for the game, I will look at the trophy list for the game and if it looks like it is platinum-able then I will usually try to get the plat. Though, that depends on how much I enjoy the game, if the game is a chore to play or has a stupidly huge amount of collectables to get to get the plat then I won't bother and basically play for the story or just stop playing depending on how I feel about the game.
I never used to be that bothered about trophies and then a friend started getting competitive about them, saying that I hadn't completed the game because I hadn't got the plat for it, which to me seemed stupid nut it pushed me to go for it. I'm not that bothered now to be fair, though I did play Terminator: Salvation because I heard it was an easy plat. While the game only took me a couple of hours to complete, the game was such a ball ache to play that I actually felt like I earned that trophy. So I guess I do care about trophies to some degree.
Also, if a game has multiplayer trophies then I won't even bother trying to platinum it because I don't have the time, patience or skill necessary to get them all, especially if one of them is "Get to level 60 in Multiplayer" and the multiplayer is awful like in 2013 Tomb Raider, that shit can fuck off. Unless they are token trophies like in Uncharted 3 where you only have to play the 2 different multiplayer modes once, then I will dabble with it once and then forget all about it.