This achievement is fairly simple to obtain -
The following Primary Weapons are available for the Tactical class;
Plasma Incinerator
Auto Bolt rifle
Bolt Rifle
Melta Rifle
Heavy Bolt Rifle
Stalker Bolt Rifle
Bolt Carbine
What I did was to create 3 loadouts in the Armouring Hall and play the second Operation's mission, 'Decapitation' on Minimal difficulty, (reason being it's currently the fastest mission and you can skip two stages). I would then use one of the required Primary Weapons until I felt confident I had accumulated 25 kills, (you can also check your achievement tracker). At the next available weapon drop pod I would then alternate to a weapon I had not yet used. It may require multiple runs!
I don't believe the Bolt Pistol and Chainsword count as primary weapons, but to be on the safe side I still ensured I had 25 kills with them both. It may also be worth mentioning, I am unsure if the Primary Weapons utilised in the trials for the Tactical class count towards this achievement.
Thank you to KrabLewis for clarifying; 'I used the achievement tracker during the trials and can confirm that it counts. I think the first and the 3rd trial use the same weapon'.