Not so much a solution - more a tip / trick to try and help from my experience.
The best way I found to approach this achievement was picking the Assault class and playing Galactic Assault. A lot of the time, the very first objective is attack / defend just one point on the map - but in all honesty it doesn't matter if there's two objectives - this still works.
Using the Assault class, when you spawn, immediately go in to Vanguard mode by clicking LB & RB simultaneously as it gives you a significant speed boost, especially as it's running out. (Most people don't seem to do this, as they want to get near the objective before going into vanguard mode to use the shot gun - whereas all we want in this scenario is the head start over our team mates.)
When around 30m away from the objective, throw your grenade forward towards the objective (preferably from behind cover) then pop out from cover already firing. Sometimes you get a grenade kill, sometimes, a kill of the players trying to avoid the splash damage of the grenade.
I could consistently get either the first kill or one of first few kills in the match this way.
Hope this at least helps someone out there.
ps. Indoor objectives work the best for this technique, as players seem to funnel in to the objective rooms, but i've done this same technique on outdoor maps successfully.