Try "Introduction to the Theory of Categories and Functors" by Bucur and Deleanu. They have a treatment of the isomorphism theorems in abelian categories beginning on page 101. They start by noting that the first isomorphism theorem follows from the definition of abelian category. Then they go on to prove the second and third theorems. Their discussion of abelian categories only begins on page 87, and they finish the third isomorphism theorem on page 111. If you already know the preliminaries, you might be able to get right to the heart of the matter without too much trouble.
Edit: I just stumbled onto a paper by Barros and Pombo, "A direct proof of Noether’s second isomorphism theorem for abelian categories". It's only 8 pages long. They reference Bucur & Deleanu, Freyd, and Grothendieck. You can find it at http://www.scielo.org.co/pdf/rcm/v43n1/v43n1a04.pdf.