Can't hear fuel pump, relay buzzes and car won't start

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My car battery died after sitting unused for about 8 days (completely dead, no dash lights, no radio). Then I had to travel and left the car for three months. Now I recharged my battery with a trickle charger, the car cranks but won't start. I've also noticed a buzzing sound with the key in "On" position, the sound comes from the engine bay, and goes away when the EFI relay is removed. I can not hear my fuel pump come on when I turn the key into "On" from the "Acc" position. I suspect it's an electrical issue - it may be corroded connections however I did clean the battery terminals and connectors a bit. Also my battery is probably not in the best shape, however if it's strong enough for a few cranks, it should be able to turn on the fuel pump I think? Please help!