This is relatively straightforward, but I will note that if you buy an augment and then sell it, it still counts towards the total so you don't need to have or equip the augment. I don't believe you even have to win the tournament for it to count.
I got this by going into one of the earlier tournaments (3 or 4) that have 3 stages after I had leveled up my robot and go through the fights with an eye towards earning cash. Just focus on overkill and chaining combos together. It also helps to use a robot with an arch-enemy (who exists in the stage). On the last stage go to all of the stores and buy and then sell the augments in order from most expensive to cheapest. This will also get you the achievement for buying all the augments from a single store.
Please note that increasing health and augment slots and healing do not count. This only applies to equipable augments.
You can also transfer cash between tournaments by buying and keeping extra augments you don't want, since those can then be sold later.