You started off right. Your goal was an implication, so you applied the introduction rule.
Then your goal was (A => B) |- ((B => C) => (A => C)).
That's another implication, so you might as well apply the introduction rule again.
Now you would have:
(A => B), (B => C) |- A => C
Here you have another implication, so you might as well apply the intro rule once more. Then you'd have:
(A => B), (B => C), A |- C
Now you can start working your way downward with elim rules.