Either have your charm or intimidate at its max, or talk to him after each mission and find out about some missing family armor that he wants recovered. Go on the side mission to recover it and he'll trust you enough to listen to you on Virmire.
Annwyn193- In order to keep Wrex alive, you must speak to him in often in the hull of the Normandy. Take the stairs, go down the elevator, he's the first teammate to the left, leaning against the wall. He will give you a mission. His grandfather charged him with the task of retrieving his family heirloom-- battle armor. After the Rachni Wars, the Krogan were not allowed weapons or armor. A turian named Tonn Actus got his hands on it. He's made lots of money selling stolen Krogan artifacts. You must take the mission and complete it before setting a course to Virmire. This will gain his trust and he will stand down, stating that he trusts Shepard enough to follow his/her lead. If you have a full charm or intimidate bar, the option will be available to you to persuade him without taking the mission, but the best way to do it is to take the mission. The only thing you lose by NOT taking the mission is experience, credits, weapons/armor that's hidden throughout the base, and various Codex entries hidden along the surface of the planet plus any minerals out there that need surveying.