I found (annoyingly) that the challenge for giving Claptrap a High Five is one time only. Before I miss any one-chance challenges later in the game, what challenges or achievements are missable inside a single playthrough?
According to Xbox360Achievements.org, there are no missable achievements. Even Up High, Down Low can be completed later on as there are other opportunities to high five Claptrap (including one right before the end of the game).
Well... there is a missable challenge called "Save the turret".
It is part of the very last main story quest and thus can only be completed once per playthrough. It may be that you can complete it afterwards if you join a coop-game with this story mission active.
Also the challenge "Compl33tionist" is partly missable. For rank 5 you need to complete 100 optional objectives. This requires 2 playthroughs but you are allowed to miss on 18 optional objectives. Also optional objectives from DLCs do count.
These are the two you should look out for.
There aren't really any missable achievements, but:
There are 4 character specific achievements that you'll have to start a new playthrough in order to get. So 3 of those will require a second playthrough.
There is an achievement for reaching level 50 and an achievement for a boss who's only defeatable at level 50. However in the first playthrough enemies are capped at level 30, and you won't get much past about level 35 killing them. The only way to get to level 50 is a new game + mode called "True Vault Hunter" where you start again and the enemies start at level 30 and go up to 50. So 2 more achievements require a second playthrough.
So there are 5 achievements that you cannot get in a single playthrough. The minimum number of playthroughs is 5 - one for each character and one on NG+ to get to level 50.
However, that's not really how Borderlands 2 works - gear can be shared between characters and badass ranks carry through. You will level up amazingly fast (about a level per kill) if you join some level 50 friends with a new level 1 character. You don't really end up playing every mission with every character and starting the story again doesn't really feel like a new playthough because you're not starting from scratch.