Honestly, this is really easy (and this coming from someone who shudders at the 'without dying' achievements!)
It's reasonably likely you'll get it the first time in the first level - the fights are all short and very easy. The only thing that might kill you is the jumps, as the jumping controls aren't great. If you do die, you can replay the first level at any point during the game (accessed from the map, same as you did in the first place).
Alternatively, just try out the second level. True there are more fights but they're not exactly challenging, and there aren't any even remotely tricky jumps. (There are a couple of tall buildings I guess you could jump off accidentally but it's really unlikely, and again, you have the option to replay levels at any time.)
One note - you can't regain health during a level. This won't really be a problem as it really is that easy, but it's worth knowing nonetheless.
Edit: As Wraitrock rightly says, you're going to get this achievement on the last level anyway, as the Tightanic Battle achievement requires you to beat the last boss without dying and the boss fight is the whole level. All you need to do for that boss fight is shoot everything he throws at you, and destroy the columns before he gets chance to hit you with them. Destroy 3 lots of columns on 3 different buildings and you'll have won this achievement and Tightanic Battle.