The "Soldier weapons" are those exclusive to the Soldier class:
-The MP43 Assault Rifle
-The Panzerschreck Rocket Launcher
-The Flammenwerfer Flamethrower
While the MP40 submachinegun and Kar98 rifle are both available to the Soldier, they do not count for this achievement.
Satchel Charge is used by pressing "Y' to toss a charge, and then again to detonate.
Veil Strike is done by pressing "Up" on the D-pad. Note that you'll need to unlock Veil Strike by purchasing it ($500) prior to being able to use it.
The kills can be over multiple sessions, this doesn't need to be done in one match.
Once you've achieved one kill with each of the above, you'll need to finish your current round for the achievement to pop.