What is the biblical basis for the priesthood of all believers? What is the biblical basis for the priesthood of all believers?
This doctrine was made evident from several passages:


Jesus is the only mediator between God and man (1 Timothy 2:5).   

Therefore, priests as mediators is against this biblical passage.
We are called a holy people, a royal priesthood (1 Peter 2:9)
Jesus blood has made us into priests and kings (Revelation 5:10)
The apostles were not supposed to be regarded as leaders, but as servants (1 Corinthians 4:1)

They were the foundation of the church because they were: entrusted with the mysteries God has revealed (1 Corinthians 4:1), not because they were given special authority


Other passages that support the idea of the universal priesthood:
Exodus 19:5, Revelations 1:4-6, and other passages in the book of Hebrews.

Wikipedia is a good source for more information.