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Some of this is just luck, but there are many other factors that can help as well.
Trying to ensure you level up quicker than most can put you at an advantage, as well as team chemistry, knowing where to position yourself, and having a high damage character with good critical builds.
Probably the best position to get this is when defending your guardian, but other instances where the teams are grouped up such as when attacking the enemy guardian or creatures may work as well.
Tripp is the character I used, and many others recommend, but any character with good damage and criticals should work. Good control characters may help as well, either while playing as or while a teammate/crew member plays as. Xenobia and Pakko come to mind for this role.
You do have a decent amount of time between each kill, so take a deep breath and pace/be careful rather than rush and mess up your chance. Just don't take too long. In mine at least, I did get the last hit it seems on each enemy, but I'm curious as to whether full/final eliminations are actually necessary for all 5.
Here is an example of my achievement earning penta kill: