An alternative method to Chimaera36's guide is to playthrough the Narco Hell : Scavangers
chapter. During this chapter you will only face enemies armed with pistols and on lower difficulties they don't pose much of a threat; allowing you to simply run around using your melee to kill them.
After completing the campaign for the first time I had only gained 46 melee kills, then after playing through this chapter twice, focusing solely on melee kills, I unlocked the achievement.
Here is a video from Youtube if you prefer a visual guide.
Credit goes to Youtube user "FireintheConsole
" for the video.