Only the stars from campaign mode count towards this, don't worry about quick battle mode. You will have to three star each of the 8 tournaments.
One star is awarded for completing the tournament.
One star is awarded for surpassing the "money donated to Kernel" amount.
One star is awarded for surpassing the overkill amount.
All three stars have to be earned during the same attempt.
Money donated to Kernel simply means leftover, unspent money. If you're still spending money on upgrades and augments, I wouldn't worry about trying to secure this.
The developers have not made it known what the minimum requirements for the stars were for each tournament. I took a screenshot of my scores for each tournament to give you an idea. However, for all I know I could have wildly surpassed the requirements, or just made it by the skin of my teeth.
Tournament 1: 8810 donated, 22 overkill
Tournament 2: 13877 donated, 44 overkill
Tournament 3: 22429 donated, 62 overkill
Tournament 4: 22122 donated, 74 overkill
Tournament 5: 23601 donated, 50 overkill (20000 donated, 47 overkill also reported)
Tournament 6: 18015 donated, 24 overkill
Tournament 7: 34128 donated, 61 overkill
Tournament 8: 55673 donated, 67 overkill
Screenshots can be found in my captures. If anyone meets the 3 star requirements with a lower score, let me know and I'll edit this accordingly!
As far as tips;
- Wait until you've progressed far enough through the game to at least have two really powerful robots (all augment slots, 40+ HP. Tournaments 1-7 only require 1 strong robots, tournament 8 requires at least 2. Ideally, you can wait until after unlocking the Super Team achievement as your robots should be plenty strong by then. You will not be hitting the 3 star requirements on your first attempt of each tournament.
- There are two types of augments; in-battle augments and map-based augments. For in-battle, I like Free Attack, Free Defense, Attack Harmony, and Attack +3. I also use Defense Harmony sometimes. For map-based augments I like Cash x3, Move+, and Axe. I also try to keep spares of the unique things like Gas Masks, etc, and equip/remove them as needed. Cash x3 easily pays for itself overtime. If multiples any map pickups by 3, turning a 300 gold pickup into 900 gold, etc. On the earlier maps one pickup can be close to 10% of the required donation amount!
- If you've got a strong enough robot with Attack Harmony (Mackenzie begins with it!), Attack +3 and Free Attack then the overkill thresholds should be fairly simple to hit. Just make sure you get into enough encounters before the CPU kills the other enemies and you miss out on the chance for overkills. With this setup you can probably kill and score +10 overkill on each robot by the time you hit 3 energy on the first few tournaments. The later tournament's enemies have higher HP, though, and require multiple attacks.
- There is a sort of "trick" you can use to secure more money donated to the kernel. Obviously, you do not want to spend any money, even on healing. The trick is that you can stockpile expensive augments from previous tournaments/attempts and bring them into you 3 star attempt. When you're sure the match is proceeding well and you have enough overkill to make it worthwhile, just hit the shop and sell all of your "extra" augments. This could be an additional ~5k-10k donated depending on how many spare augments you brought.
Tournament 8 is a bit different than the others. It has 3 maps, and has you use your entire team. Map 1 randomly selects 4 of your robots. Map 2 uses the remaining 4 robots. Map 3 uses all 8 of your robots (or however many survived the previous 2 maps. This is why it is important to have at least 2 strong robots; you can keep restarting Tournament 8 until your sure that one robot appears on Map 1 and the other on Map 2. Obviously, though, it's best to go in with 8 strong robots after completing the Super Team achievement.
The first two maps have 4 fairly weak enemies each, and a bunch of environmental hazards. You can get a lot of overkill across these two maps, and will want to secure the overkill amount here. Use the aforementioned Attack Harmony setup to help.
The last map only has one enemy, the final boss, that has 100hp. You will not be getting much overkill from him, so this is why it's important to go all out on overkill on the previous two maps. You could theoretically just throw all of your robots at the boss until you manage to kill him, but if a robot dies I assume that you lose all of the money they collected. I managed to defeat the boss each time I played through with a single robot. This battle, though, I did not use the Attack Harmony setup. Since the boss has 100hp I figured it'd be more of a battle of attrition and focused on defense using Defense Harmony. My setup was Free Attack, Free Defense, Defense Harmony, Attack +3. I'd focus on keeping my defense up to be sure I could outlast him and only attacking when I knew I was safe. If you save up 4 attacks you can attack twice in one round. Once to deplete his shield and once to do pure damage.