Execute an Auto-Combo with Groot.The game never explains what Auto-Combos are, but they're relatively simple: during combat, if you melee () attack an enemy the same time an ally melee attacks them, it will initiate an Auto-Combo with that ally. So for example, if you melee an enemy the same time that Gamora is hitting them with her sword, a very quick scene will play of the two of you attacking that enemy together.
So for all four of these achievements, you need to melee an enemy the same time as that ally. This is particularly easy to do with Drax and Gamora because they are entirely melee fighters. Simply follow them around and attack the same enemy they do. Groot sometimes hangs back and uses ranged attacks, and sometimes melees, so you'll need to pay attention to when he's using melee on an enemy and take that opportunity to melee them at the same time.
Rocket, on the other hand, is virtually entirely ranged attacks and rarely gets up close to enemies. Pay particular attention to Rocket during fights and take advantage of any time he is up close with enemies so you can unlock his respective Auto-Combo achievement. Since he tends to hang out on Groot's back, you first need him to get off Groot's back. This will happen automatically if an enemy hits them, but I also had some success with using some of Groot's abilities, which would sometimes force Rocket to jump off. Once Rocket is off, you need him near an enemy (because otherwise he will just hang back shooting his gun). There are two ways to do this: either stand right next to him and wait for enemies to come to you (since enemies will always gravitate toward you over your allies), or once you unlock the wind elemental ability, you can use it to pull an enemy toward you and Rocket. This stuns the enemy though and I never had success with it. Either way, once you get an enemy next to you and Rocket, hover over and be ready to melee as soon as Rocket stops shooting. This is definitely the toughest one and can be annoying, so make it a point to get this one out of the way as soon as possible.