I'm sure you could boost this in a Private match, but in a public match I found the best way to get it was:
Select the mode 'Free For All'
Select the Type 'Casual'
When the game starts, just keep looking for opportunities to use the 'Charged Melee' attack, which you instigate by holding Y.
Doing so causes your character to sprint with their melee weapon held high and connecting with an enemy is an instant kill.
Look out for enemies with their back to you and even consider doing charged melee's around corners on the chance that you'll connect with someone coming the other way.
I can't be certain but it also felt like Charged Melee enables you to run longer than the sprint button does!
ADDITIONAL TIPS FROM THE COMMENTS:
RainbowPooper has added a useful tip: "In rescue you can finish downed enemies (if you are the Solarii) with Y when you are close to them. It will count towards melee kills too."
NOBLEPX672 has added that Rescue may be the best mode to try as you actually have to kill the enemy with a melee attack.