What should be done is a combination of both cartoons, SSP and TAS. Say ASM 3 starts with the announcement that John Jameson has discovered a unique new life form in space. The symbiote will be brought to Empire State for staff (Peter included) to research. While being studied, it could be known that this organism feeds off emotion. Then one night while Peter is alone in the lab, Green Goblin attacks, and the lab is destroyed, but in the fight, the symbiote attaches itself to Peter. It's a simple storyline, and even a complete idiot could've done it right the first try.
Symbiote makes Peter faster and stronger, but also has a negative effect on him.
Peter gets Brock fired.
Brock gets mad at Peter and/or Spider-Man.
Gwen shows concern over Peter, and it gets to him.
Green Goblin is still terrorizing, but with the suit, Peter is menacing toward Norman.
Then basically go from there. Eventually have Peter go too far, forcing him to disband the suit. He loses the suit, and Brock gets it. Brock and the symbiote's hate merge to create Venom.