This achievement is not for melee kills but for executions. You execute an enemy by holding Y when you see a skull icon above an enemy. Some unarmoured enemies will have this straightaway. You will know when you have performed an execution as you will get a short cinematic showing you gutting or beheading the unfortunate individual on the wrong end of your skyhook.
More heavily-armoured enemies will need to be damaged before the skull icon appears. A basic melee attack closely followed by an execution works well with the lightly-armoured enemies.
If you're patient enough, you can grind this early on. Reloading a checkpoint does not affect your cumulative total (for this or other "do something a certain number of times" achievements - you can use this to your advantage to get a few kills shortly after a checkpoint then simply reload the checkpoint. Rinse and repeat until you get to 20 kills.