It does affect the way I play in some cases. But only for games I really really like and if the achievements are not stupid.
For example I love the new XCOM game and it has some of the best achievements I've seen, with a few stupid in between like "kill x amount of aliens". I only miss the two multiplayer achievements since I stick to single player only.
I liked some of the achievements for the Witcher 3, since it's another game I love. And it's definitely the type of game where I'm interested in achievements. I mean, I have right now 112 hours in the game, in a single playthrough, and a lot more left to do. I'm going so slow because I want to explore/try everything this game has to offer, complete it as close to 100% as possible (I have one failed quest atm). Trying to do the achievements, especially some more challenging ones really appealed to me but then the latest patch broke them and you can't get any more achievements. Pissed me off to no end, especially since I though I completed a few of them, before finding out the system is broken.
And if it's a game I like I try to get the rarest achievements.