What is the Biblical basis for multiple judgments after death?

It sounds like you're referring to the White Throne Judgment and the Bema Seat Judgment, which are typically believed to be different judgments by Evangelical Protestants.

The White Throne Judgment is described in Revelation 20:11-15. This, to the traditions that believe in the two judgments, is the judgment that everyone faces, where you are judged to be innocent or guilty; whether you will be cast into the eternal lake of fire, or accepted into Heaven.

Revelation 11-15: (KJV) 11 ¶ And I saw a great white throne, and him that sat on it, from whose face the earth and the heaven fled away; and there was found no place for them. 12 And I saw the dead, small and great, stand before God; and the books were opened: and another book was opened, which is the book of life: and the dead were judged out of those things which were written in the books, according to their works. 13 And the sea gave up the dead which were in it; and death and hell delivered up the dead which were in them: and they were judged every man according to their works. 14 And death and hell were cast into the lake of fire. This is the second death. 15 And whosoever was not found written in the book of life was cast into the lake of fire.

The Bema Seat Judgment, on the other hand, is believed to be the judgment of rewards to those who are saved, who have in essence, "made it through" the White Throne judgment. It's based on several scriptures scattered throughout the Bible.

There's an article that describes it here, with Scripture references. The page has a copyright notice, so I won't copy sections of it, rather a short summary follows:

Several passages in Scripture indicate that we will receive rewards for faithful service. - 1 Thessalonians 2:19-20 - Revelation 22:12 - Romans 14:10-11 - 1 Corinthians 3:11-15 - 2 Corinthians. 5:9-10 - 1 John 2:28 - Revelation 3:11-12

Both Romans 14:10 and 2 Corinthians 5:9 reference a "judgment seat", which is a translation of the Greek word "Bema". There are arguments from some circles that this term has connotations of rewards coming to athletes competing for a prize. The article I referenced above states it this way. Others argue that these explanations are extra-Biblical and contradictory.

Whether or not there are separate judgments is something that's been debated for centuries. I won't put my own personal beliefs here, as they are flavored heavily by the doctrines I've been immersed in.

However, I will say that from an evangelical protestant perspective, the doctrine of two separate judgments does reconcile otherwise contradictory passages.

• First, the doctrine of salvation by faith alone apart from works would indicate that our good works do absolutely nothing to earn us salvation. (The White Throne Judgment)
• Second, we are told in various places in Scripture that we will be judged by our works, and will be rewarded. If the core doctrine of salvation by faith alone, apart from works is true, then these judgments and rewards must be something other than salvation.
  • There is the promise of crowns in Heaven (Rev. 3:11, Matt. 6:2, Rev. 4:4)
  • There is the promise of Heavenly treasure (Matt. 6:20, 1 Pet. 1:4)
  • There are promises of Responsibility, Authority, and inheritance of God's possessions. (Matt. 19:28; 24:45-47; 25:21, 23; Lk. 19:17-19; 22:29-30; Rev. 2:26).

As to whether the doctrine of two judgments is Biblical depends on the definition of "Biblical". Scripture does not come out and state directly that there are two judgments, so in that regard, it might be considered "extra-Biblical". On the other hand, there is a bit of Biblical support for the doctrine. In that regards, it's as Biblical as the doctrine of the Trinity, and other doctrines not explicitly stated in Scripture.