mI'm not the best DK by any means, but my combo game is up there. So i'll gladly post some things that should help
This would literally be the longest post I've ever made if I was to get deep into his combo game, so instead I will list some small things like what his best combo moves are and stuff so you can experiment yourself to see what fits you. DK's combo game is one of the best in PM, so there is no right or wrong way to combo. He can combo in many ways.
Out of a grab (assuming cargo u throw won't work):
Experiment with d throw and standing cargo f/b throw follow ups with different DI. Some follow ups for d throw are regrab if they DI toward you, dash attack, nair, an instant up air, and tech chase with down b, (I actually do a lot of techchasing with down b in both of my combo videos, so maybe watch those. I will link them at the end of this post. There is a lot of funny and janky stuff in those videos but there is a lot of useful stuff as well lol)
When you want to combo out of cargo f/b throw, most of the time you should throw them toward the middle of the stage so you can follow up with a dash shffl nair into another dash nair or fair for example (also in my 2nd combo video, cargo b throw, nair, fair). If you throw them towards the edge, you have less space to dash and follow up without running off stage, so it's harder to extend the combo. However, if they starting DI'ing towards center stage, you should then throw them to the edge to get the free follow up.
Safe moves in neutral game for starting combos/follow ups:
jab combo: often can combo into itself or a grab. It's safe because if you miss the first hit, you can dash away for example
d tilt: Can't really truly combo out of it, but d tilt to dash Jump cancelled grab will work 90% of the time.
bair: bair before landing into turn around dash jump cancelled grab works as well
nair: nair approach has many good follow ups(grab, d tilt to grab, up tilt, jab combo)
The reason I posted "safe" combo starter moves is because in theory, you can follow up with any move in the right circumstances. DK is combo food so it's important to play safe with him and get hit as little as possible.
Lastly, if you want to improve on extending your combo game I recommend you read through this list of GREAT moves for continuing juggles and find out which one(s) you should incorporate more
-up air
-up tilt
-nair(weak and strong hit)
-bair(weak and strong hit)
-jab combo (situational)
-dash attack( if they DI away after a shffl nair at combo percent, I almost always dash attack because it's the only move that will hit them before they land v.s many characters)
Example of applying everything in this post

for fun lol)
on the left side of FD facing left V.S. a middle-ish weight character and assuming they DI as I expect: Jab combo(possibly 2), grab, cargo b throw, shffl nair, dash attack, short hop up air, full hop fair
Btw that combo was not just speculation and neither was anything in this whole post. I actually did almost that exact combo yesterday. This stuff really works!
Link to my combo videos:
https://www.youtube.com/watch?v=E4vmJ2KrYZ8
https://www.youtube.com/watch?v=FU48YZfKp8Q
Edit: ignore the double side b's in those combo videos lmao. Also, the second link is much better as far as having things that would actually work more often, not including the last minute or so
