I was able to do this while playing solo and without any other guides.
To control the altitude of the Phaeton, on the controller setup that I play I had to use
I noticed that after killing 2 of the initial Banshees that are flying around the Kraken, carefully hitting them with the Phaeton's guided missiles from an angle where I wouldn't hit any of the Kraken's turrets by accident, a big passage opens at the bottom front of the Kraken and 2 more Banshees come out. You can take advantage of this window and rush inside it, destroy the core with sticky grenades or some other power weapon and then get out of the Kraken.
It is a fast get in and get out thing to do and I had no problems with my team destroying any turrets while I did that, so the achievement unlocked.
If this idea helps you in any way, feel free to leave a thumbs up. Thanks.