Is the doctrine of "by faith alone you're save" Biblical?
You should first define what you mean by “saved.” We are justified by grace alone through faith alone.
Did God, the Prophets or Jesus taught such doctrine?
Jesus did:
Jhn 3:14 And as Moses lifted up the serpent in the wilderness, so must the Son of Man be lifted up,
Jhn 3:15 that whoever believes in him may have eternal life.
Jhn 3:16 “For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life.
Jhn 3:17 For God did not send his Son into the world to condemn the world, but in order that the world might be saved through him.
Jhn 3:18 Whoever believes in him is not condemned, but whoever does not believe is condemned already, because he has not believed in the name of the only Son of God. (ESV)
And Paul did:
Eph 2:8 For by grace you have been saved through faith. And this is not your own doing; it is the gift of God,
Eph 2:9 not a result of works, so that no one may boast.
Eph 2:10 For we are his workmanship, created in Christ Jesus for good works, which God prepared beforehand, that we should walk in them. (ESV)
So, both Jesus and Paul, which means God did as well.
Works don’t save anyone, otherwise the Law would have been all we needed. Works are only evidence of our faith. If someone professes to be a follower of Christ isn’t doing good works and, especially, isn’t that interested in doing good works, then their salvation is suspect.
Believers will still be judged for their works but to determine reward, not salvation.
1Co 3:12 Now if anyone builds on the foundation with gold, silver, precious stones, wood, hay, straw—
1Co 3:13 each one's work will become manifest, for the Day will disclose it, because it will be revealed by fire, and the fire will test what sort of work each one has done.
1Co 3:14 If the work that anyone has built on the foundation survives, he will receive a reward.
1Co 3:15 If anyone's work is burned up, he will suffer loss, though he himself will be saved, but only as through fire. (ESV)
2Co 5:10 For we must all appear before the judgment seat of Christ, so that each one may receive what is due for what he has done in the body, whether good or evil. (ESV)
Also, Matt 25:14-30.