Well come on, she can't stay angry at Zuko forever, because then the situation would be like this:
Katara: Thanks Zuko for teaching Aang firebending and helping him defeat the firelord along with us!... But I still hate you.
Zuko: WAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAHHHHH!!!
Yeah, pretty much. Anyways I think there are three ways he could gain her trust:
1. Not harm Aang or be in the wrong place at the wrong time for 5 episodes.
2. He could save Aang's life or another person's life in the group.
3. (This techniacly falls under the category of theory #2) He could save Katara's life.
What would really gain her trust is saving her life. Because without her in the way then Zuko wouldn't have to worry about "taking a step backwards" in Katara's point of view. So theory #3 is my best bet.