I was reading through Ravi Vakil's book/lecture notes on spectral sequences, but I came to an impasse. He leaves as an exercise the construction of the $d_2$ differentials of the spectral sequence (associated with a double complex). I know how to construct it using the big bad machinery of filtered complexes, but in the notes, he gives a hint that there is a more elementary proof that is pretty similar to the proof of the Snake lemma.

It's not clear to me how a diagram chase could really help though, since I don't see any short exact or even long exact sequences, which are used extensively in the proof of the Snake lemma.

Am I missing something? Could someone fill in some of the details?