Deliver a sample of jet to Dr. Troy

You must first be a LEGIT citizen of Vault city (either passing the test at the council chamber or solving the Gecko power plant problem).

Find a sample of Jet, either buying them from any drug dealers in the game or pick one up from a dead highway man/raider in your travels from The Den to Vault City.

The vault is located on the right side in the first part of the city (after the "outsider zone").

Talk to Dr. Troy inside the Vault.