Advice from someone who occasionally dabbles in pvp but is not very sweaty -

So the three sub-achievements for this are:

Emperor Slayer: This can be a pain and you just kind of have to get lucky I suppose. You need to find an enemy emperor (it does not have to be *the* emperor of the campaign you're in, just an enemy player who's currently emperor in one of the campaigns - I was told it also counts if you kill them in IC while they're currently emperor in a cyro campaign) and (at least as it was explained to me) actually get the killing blow.

But you can improve your odds by joining a group in cyro when it's busy and your faction does not currently have emperor, and following the action. The enemy emperor can often be found with a group in the thick of the action, taking keeps, capturing and running scrolls, etc. You may be able to tag along with your faction and get lucky with a kill shot, particularly with a good execute skill like templar beam. (Even if you can't actually get *into* your faction's active groups, you can try to follow them).

Alternately, if you're not really good at pvp and don't want to jump into the thick of things, getting involved in defending keeps against sieges can be one way to do this. I got emperor slayer without even realizing it because I happened to hit the emperor with a siege weapon. This may also be easier during mayhem with the extra campaigns active.

The Sublime Protector: IMO this is actually the easiest one if you aren't into pvp, since it just requires doing the IC zone quests. You also get a nice dye. The quests are pretty quick and easy as long as you don't get ganked. I did them at a quiet time, in no-cp IC, on a nightblade so I could remain cloaked as much as possible. There are a few parts where you have to accomplish a PvE task that will leave you exposed to attack, but hopefully no one will notice you if it's quiet. I only encountered one enemy player the entire time... they started attacking me but I just stood there blocking and I think they realized I was on the quest... they just walked away. Alternately, you could wait for a busier time when your faction is very active in IC so that people will be around to potentially help you if a ganker shows up.

Conquering Hero: For this you have to win 50 BG's. Just gotta do them, you'll win some and lose some. It's really unbalanced right now, be prepared to be on some bad teams where you'll get stomped. Just gotta shrug it off. You'll also wind up on some good teams and hopefully improve to where you feel like you're contributing and maybe have a little fun. Eventually you'll get to 50 wins just by participating though.