God, through Moses, warned Israel that all the firstborn in the land of Egypt shall die (Exodus 11:5), and that would have included Israelite firstborn, too. The threat against the Israelites was real, and they had to prove to God, through the sign of the blood on the doorposts, that they wanted to be separate from the Egyptians. Without the blood, the Israelite firstborn would have shared in the same judgment as the Egyptian firstborn.
The blood on the doorposts represents the life of the lamb given to redeem those within each participating house (see Exodus 13:13-16; 34:19-20), not to symbolize forgiveness. God does not draw attention to the Israelites' sins in His Passover instructions, even though the Israelites were sinning - grievously, in fact. Through Ezekiel, God says that the Israelites as a whole were unabashed idolaters at this time, and God nearly destroyed them, then and there:
