I can confirm that this only requires a 20 elimination streak, not a 20 kill streak. So you need 20 kills and assists for this achievement.
As for how to get that, I would recommend any character that you are good with, but flanks are likely the easiest way to get this. I used Skye and got close to a streak of 30 when I got the achievement. She is a very easy character to use and is available for free to use for all players. Other than that, it's all luck in getting a good team, facing a mediocre one, and having the match last long enough for you to get enough eliminations.
EDIT: This can not be done against bots. It must be done in an online match against real players.
EDIT 2: This might be able to be done against bots. This achievement seem to simply have inconsistent unlock conditions.
EDIT 3: Apparently the game has been updated such that achievements are now unlockable in training matches against the bots. This makes the achievement much easier than doing it against real players.