As a true-to-the-formula third person shooter you should know what to do to get kills whether by bullets or grenades. Fuse also provides tell-tale red energy bars above enemies, letting you know how much more damage they can take before you get the kill.
Alternatively hammer whilst next to an enemy to melee them to an early grave or press when prompted to perform a slick takedown.
Echelon mode is best if you are grinding for kills specifically.
Keep track of your kills by going to Service Records in the Pause menu. Go to Leaderboards and press right on the D-pad twice. This will show your total kills.
The ‘All’ menu in Services Records will only show you your Campaign and Echelon stats separately.