The easiest way to do this mission is to complete Wrex's "loyalty" mission. To do this talk to Wrex after a main story mission (Feros, Noveria, and Terrum) and he'll eventually talk about his family's armor. Investigate him and he'll give you a location where you'll have to clear a room of pirates. Bring a character with high enough decryption to open the safe in the upstairs room. You can either have Wrex in your party for this mission or immediately after you return to the Normandy talk to Wrex. He'll talk about how he's starting to like you.
Later once you're on Virmire (choose to go there only after completing that loyalty mission) you'll be given the option to reason with Wrex because you supported him and he'll agree that he'll trust you in this situation.
Alternately you can have a high enough Charm/Intimidate score to convince Wrex instead, however this has the possibility of being not high enough on your first playthrough when you reach Virmire as a point value of 8 is necessary.