Vanquish

The most important word in the achievement's description, is "only". An easy Act to get this achievement on, is Act 2-1 (River) on casual difficulty, but there will be other opportunities later in the game as well. On Act 2-1, you'll come out in the open where you will find 2 Romanovs.

Get close to one, melee attack it with , and seek cover with as your ARS temperature gauge will indicate it's overheated. Solty has suggested not to use the disc launcher, as it will launch a disc at the end of the melee attack, and thus it won't count as a melee. Boost-melee seems to be quite destructive! Rinse and repeat these steps. Focus on one Romanov at a time. But never shoot at the Romanov, only melee attacks. It's alright if your NPC teammates shoot at it, as long as you don't!! If you accidentally shot one, press and reload the last checkpoint, then try again. On casual difficulty this shouldn't be a challenge at all, as you almost seem invulnerable.

Since you can only use melee attacks to get this achievement, it will also unlock the "Romanov This!" one.