Needs More Whammy Bar

There are a few things that you have to keep in mind when going for this achievement:

Have the "Prophet Birthday" skull enabled.
This is pretty obvious just from the achievement description, but remember to enable the skull. You can have other skulls activated as well, if you want.

Play the level in Anniversary mode
This skull/achievement/Easter egg doesn't exist in the original game, and if you play on original graphics you won't experience it. I haven't confirmed whether or not this affects the achievement, but I would assume so.

Kill Regret by punching him
This isn't the only way to kill the Prophet, of course, but its the only one that triggers the Easter egg. Again, I'm not sure if this affects the achievement, but I would assume so.

What you need to do is approach the Prophet in his hover chair thing, press the Use button to climb onto it, and press the Melee button repeatedly to beat the snot out of the Prophet. If you've followed the previous steps, you should hear a guitar riff every time you punch the Prophet.

After the closing cinematic ends, you should receive the achievement!

If you, like me, did the above and didn't get the achievement, you will have to do one additional step.

Play through the level Delta Halo with the Prophet Birthday skull enabled

I am not privy to the technical reasons as for why you have to do this to trigger the achievement, but I can confirm that it works if the above didn't do the trick.

Once you finish the level, you'll receive the achievement. Congratulations!