This doctrine was made evident from sevearlseveral 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.