Cole and Shaw were given the social security numbers by their handler Wilson of people who were about to commit acts of terror against the United State of America, and they neutralized those threats. He was mainly confined to the technical side of the missions like surveillance, hacking, etc. while Shaw was the field agent. They made a successful team and were very fond and protective of each other.
Cole was disturbed by one of their missions and began to question where they got these social security numbers. He was convinced that Daniel Aquino, a number they had executed, was innocent and framed by his own government.
However his inquiry into Aquino's death led Wilson to send a hit squad to take them out. He died protecting Shaw from gunfire, telling her he wanted to be her hero. He was the only person Shaw ever cared about and she was visibly disturbed by his death.
Shaw later shot and killed Wilson thus avenging Cole's death. (“Relevance”)
Following Cole's death, he is framed for having been part of a militant group and while Shaw wants to tell his parents the truth, she is aware its too dangerous. Instead, Finch hacks into the CIA and inserts him as an agent who died undercover before leaking it to the press. (“Trojan Horse”)