Since there is a limited amount of skills and upgrades you can get each round, there is a few tricks you want to use.
Only pick up one additional weapon and one passive upgrade, so you have a total of 3 upgradeable gadgets.
If a drill drops 3 items that will give you another weapon, dont pick it up, just ignore it (dont worry, you can do it withouth em when you have upgraded your character in the workshop a bit)
Try to upgrade your weapons eavenly, because there is no point in upgrading a weapon further than the special upgrade (at least not for this Achievement)
You can ignore maximum 5 Drills (if you dont get an upgrade for a weapon you dont want to max) to still achieve this achievement, if a sixth drill drops something that isnt an upgrade for a weapon you have, you wont be able to get this achievement in this run.
Reason:
You need to get 3 base weapons, 6 upgrades each, and the sevenths is the special upgade.
Meaning:
3*8 -1 (the minus because you start with one weapon)
So you need 23 upgrades
You get maximum 24 choices with drills, and 2 bosses who drop 2 upgrades each for weapons you already own.
so 28 possible upgrades, meaning 5 "spare" for a weapon of not your choice.
The bosses only drop upgrades for weapons you have, which is why i recommend only owning the 3 gadgets you want to level to the special upgrade.
Video example here:
(and yes, i accidentally walked on to the homing missiles, i got very lucky to get the last special upgrade in the last drill)
General advise:
1. Quickly run from drill to drill
Because killing enemies will not grant you any advantage, so there is no point in wasting your time with them and risk running out of fuel or dying
2. Use the Manual expanding trigger, to control how many enemies you will release
3. Use the falling drills to kill as many enemies as you can, if you are lucky you can even smash the bosses