It depends a whole lot on a game. Sometimes when I have completed a playthrough, I take a look at what achievement I have gotten. I then browse through the list of the ones I have yet to achieve, and consider whether or not I would have fun trying get these achievements? From there's it's a simple decision of either starting to hunt down achievements or simply ignoring them. Hunting down achievements can become almost like a minigame of its own to me, if the achievements are well done and promote styles of game play that I would otherwise ignore.
However, a lot of companies don't put a lot of effort into introducing good achievements, so for the most part I end up ignoring them.