Alpha Protocol

Save manually once you're fighting him. In the convo following his defeat, choose to execute him, and this acheivement is yours. Reload your save, defeat him again, choose to spare him, and you'll get the "A Price On Mercy" acheivement.

Boss is much easier to beat if you :
- did Taipei first : buy intel from Steven Heck that will mess with his drugs. If you're stuck at the boss fight without having done Taipei, load last safehouse checkpoint, and in the mission select screen, press RT then LT, and select Taipei safehouse. Try Moscow again after befriending Steven Heck and buying his intel.
- chose the data over saving Albatross/Sie (unless you're playing veteran, in which case you can do both), since 100% intel on Brayko gives you a +5% damage perk on him, while saving Albatros/Sie gives you a 20% endurance recharge rate perk.
- use a flashbang (equip a few of them before starting the mission) when he takes his wiff (or when he's running after you) and take that opportunity (he'll be stunned) to shoot him so as to drain his resistance away (the blue bar above his health bar), then headshot him with a maxed out crack shot (pistol skill). Pistol is the best anti-boss weapon. So be sure to level your pistol skills as much as possible.

As explained here already, this achievement is for killing Konstantin Brayko, this doesn't mean beating him in the battle but for choosing to execute him during the cut scene after. It is during the Moscow missions and you'll know when the battle begins as he will make a rock star-esque entrance on a dance stage.

The Battle:

Brayko is yet again another boss that if you don't know the way to beat him, can get pretty hard. When he appears on the stage, take cover and shoot him as normal. I'd advise not getting too close to him as his weapons do little damage at range but alot at close range. The main change in this battle is when he takes his drugs (you'll see him stop and take a good ol' wiff), then he'll come straight for you. In my opinion the best way here is to run around the cover just around the stage, keeping your distance and not fighting him, he'll just cut you up pretty bad if you do. After awhile running away from him, he'll get tired, shoot him up and he will dissapear. Henchmen will come from the sides of the hall, take them out and Brayko will rejoin the fight, keep your distance again and shoot him up. The fight basically repeats from shooting him, running from him and killing his henchmen, rinse and repeat to kill him.

Execution:

Very simple, once defeating him you will have an option in the cut scene to execute him. You can talk to him for a bit first to gain information but as long as you choose to execute him before choosing to team up with him you will get the achievement.

I hope this helps guys.