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
Wikipedia is a good source for more information.
The priesthood of all believers is the most ignored, yet one of the most fundamental doctrines of the Bible. I believe this because of a basic misunderstanding of the qualifications, appointment, and function of a priest. In the 21st Century, a priest is associated with advanced seminary degrees, formal appointment as a clergy and conducting sacraments.
Arthur Lindsley observed in The Priesthood of All Believers:
The priesthood of all believers is based on the clear teaching of Scripture. The Old Testament anticipated this teaching when God said to his people at Sinai, “You shall be to me a kingdom of priests and a holy nation.” Isaiah says that a time will come when “You shall be called the priests of the Lord, they shall speak to you as the ministers of our God.”
As you can see from this quote, priesthood of all believers was a promise first made to the Israelites during the exodus. God promised He will make them a kingdom of priests, a holy nation and His own possession (Exodus 19:5-6). But there was one condition (Exodus 19:5), "if you will indeed obey My voice and keep My covenant."
As the story of Exodus progresses, we all know Israel failed to meet the condition and in the end Levi was appointed to be a tribe were the priests would come from. But, this highlights another important aspect of priesthood: priests followed a specific bloodline. Only people in a specific family were allowed to be priests.
Jesus' priesthood was of the order of Melchizedek and as believers we're his siblings. So, then God fulfills His promise of making a kingdom of priests and a holy nation in us. Uche Anizor and Hank Voss, makes this point, priesthood of all believers, clear by recounting Mark 12 when Jesus and His disciples were accused of breaking the Sabbath:
Only the priests were allowed to work on the sabbath, since their work was temple work. Second, only priests were permitted to eat the consecrated bread. Third, Jesus and his disciples were not in the temple but in a field. It is possible that what Mark (and Jesus) is conveying is that Jesus as the royal High Priest is himself the location of the sacred space. Therefore, his disciples, being bound to him, are given rights as priests to work on the sabbath and as priests are also allowed to eat in that holy place, as David and his men did.
I hope this answers your question.