Complete Act 3 on Hardcore.
The above achievements will unlock once you complete each act, there are several parts to each act so don't be surprised if the achievements don't unlock when you complete the first level of each new section.

This game has some very frustrating parts so be sure to always terraform the ground to make a shield. Try to keep your distance from most enemies as being out in the open will result in a quick death.

Fighting Hydras and Large enemies
These enemies can be quite tricky at times due to their size and agility. I suggest you keep moving, use the double jump (when acquired) and the Black-Widow/Machine gun. These enemies will take a lot of shots so standing still will just leave you open for an easy death.