How to define the natural map on the second page of a spectral sequence?

Question

I'm learning about spectral sequences in Ravi Vakil's notes, and can't quite figure out how to define the map ($d_2$) on the bottom of page 59 (he describes it as a worthwhile exercise). It should be a morphism of the cohomology complexes on page $E_1$ of the sequence. I tried using short exact sequences to get a connecting homomorphism, but I haven't been able to do it.