If you don't have the Captain Ahab maxxed out yet, I suggest you go grind it out.
Using any of the guides here will make things easier:
A small note, I use the term challenge below, that's what they call the missions/side-quests that have no story behind them. So hopefully that doesn't confuse you with the challengre achievement this solution is for.
You will manually have to travel to the Episode 4 - Buck Stops Here challenge as the fast travel for only this in-game challenge is glitched and takes you somewhere else.
Once there, start it up.
Just like Buck's other challenges the pattern is always the same per challenge.
This specific ones are:
1. Melee kills
2. Grind kills
3. Bounce kills
4. Winger kills
5. Herker kills
6. Catapult kills
Three weapons maxed out I feel are pretty useful for getting through this fast:
Hair Spray Bomb
And of coarse you'll need the Captain Ahab
Start by using the Hair Spray Bomb to build up to level two overdrive meter to activate your melee amp which should help you get through the first phase easy enough.
Next jump up on the wires and keep spamming the Hair Spray, then transition to jumping on something.
You should get those three done pretty fast.
Next the Wingers come out, this is why I bring with me Harry but any weapon you like can be used in place.
Then Herkers, pull out the Captain. They go down with three hits for me, you don't really need any specific amp on the gun. You can put the one that stuns them but they don't really move around too much during this challenge.
Then finish up this last part (those of you who have never done this before, look for the yellow spots and stand in them to lure OD, then press Y to have bombs launched at your spot, may be required to do twice to finish this step).
After you complete a full cycle everything starts over but with larger requirement to continue.
The important waves to note though are the Winger/Herker ones as they increases by one each cycle. This is a good thing as it's possible to get all five that you need in just one two minute try although if you don't make it to the third Herker cycle you can just click retry and get to the second cycle again.
MAKE SURE TO CLICK RETRY OTHERWISE IT WILL TELEPORT YOU AWAY FROM WHERE THE CHALLENGE STARTS!