Depends greatly on the game.
I haven't 100%ed Elder Scrolls but that is a time sink, i believe it has missables and requires multiple playthrus too(or at least aligning with multiple factions?)
There are games that are straightforward and just require playing thru the story, there are games that require performing specific highly detailed tasks. there are games with specific events that only happen at certain times (in game or corresponding with real world dates/times)
Before i start a game the first thing i do is check for unobtainables, missables or buggy achievements. This obviously doesn't work so well with new titles or anything that has recently been patched but i rarely play new games and always try to dodge patches.
Other than that it heavily depends on the game.
Some i blaze thru with a guide (Beyond Eyes[sucked] and Albedo: Eyes from Outer Space[decent] both started/completed today)
Others i enjoy and go back for mop up later (Pacman and the Ghostly Adventures 2 and The Lego Movie Games with my son today)
Some i do a mix. Making sure i don't miss certain achievements that would require a full playthru and/or could become permanently locked and/or are significantly easier to obtain at certain points (Marvel Puzzle Quest: Dark Reign and Killer Instinct also both played today)
It is only moderately different from how i played before achievements. Trying to see all the endings in an RPG or hit the 100% mark (or higher) in a platformer, beating the best times in a racer, etc
One difference is now i tend to be more driven to do those things. There is easily seen evidence that they were done versus the save files i have hanging out on memory cards. I don't need to keep a list of my feats or things to go back and do because my gamertag has it all there already.
Another difference is i'm less likely to play a AAA title or 40+ hour game. But if anyone ever read thru all my posts in these "games played" threads they would know i played a TON of garbage well before there were achievements for doing so