You encounter Belyash during the mission "The Show Must go on" as you reach the Theatre. The Theatre entrance looks suspiciously like a boss arena and this is where Belyash will spawn.
To get the Chop Chop Chop achievement to pop pop pop you'll need to complete this boss fight with Melee weapons only. Using any form of elemental damage or firearm seems to void this achievement, so suggest using melee throughout.
Thankfully Belyash is a relatively easy boss fight. He has slow obvious melee attacks and lunges, the only thing that might catch you out is the ground pound similar to what Hedgie used. Just jump over the orange shockwaves to avoid damage. Once Belyash drops down to 50% health he'll start using a flamethrower attack that also sets the grass on fire. While he's spraying try to get behind him to land a few blows. I found charging a heavy attack and hitting him when he lunged worked well for me throughout the fight.
Once he goes down, you should get both achievements. If not, load a previous save to try again and hopefully it pops the second time around!