You will unlock this achievement when you complete the second mission called "Vigilantism".
This level is much more intense then level 1. It has more fighting and the enemies are stronger. The platforming is also more advanced.
Overall this is pretty much a easy level and you should have no issues.
The last boss fight is fairly simple. Just kill the guy with the green shirt and red hair. If you are having problems you can take out all the other enemies first then take out the boss.
Keep in mind you are not playing as Raph but your playing as the Nightwatcher. So you will not get the "Used Raphs special move" achievement. The first time you can get that achievement is in Mission 11 called "O brother where art thou".
I have attached a walk through of level 2 if you need it.