Priesthood Requirements - Exodus 29

The Bible requirements for becoming a priest are found in Exodus 29.  Please read the chapter for more details.
Priesthood Requirements - Exodus 29
1. Select one young bull and two rams that are without blemish.
2. Take unleavened bread, unleavened cakes mixed with oil and wafers anointed with oil. 
3.  Put the food in a basket.  Bring the animals and food to the door of the tabernacle.
4.  The priests must be washed with water at the door of the tabernacle.
5.  The high priest must put on the tunic, the robe of the ephod, the ephod, the breastplate and the intricately woven band of the ephod.  Then the high priest must put on the turban and the holy crown of the turban.
6.  Take the anointing oil and pour it on the high priest's head and anoint him.
7.  Put tunics on the other priests.  Gird them with sashes and put hats on them.
8.  Bring the bull before the tabernacle and have all the priests put their hands on the head of the bull.
9.  Kill the bull by the door of the tabernacle.
10.  Take some of the blood of the bull and put it on the horns of the alter with your finger.
11.  Pour all of the blood beside the base of the alter.
12.  Take all the fat that covers the entrails, the fatty lobe attached to the liver and the two kidneys and the fat that is on them and burn all of it on the alter.
13. Take the flesh of the bull, with its skin and it's offal and burn it outside the camp as a sin offering.
14.  The priest must put their hands on the head of one ram.
15.  Kill the ram and sprinkle it's blood all around the alter.
16.  Cut the ram in pieces, wash the entrails and legs and then put it back together with its head.
17.  Burn the whole ram on the alter as a burnt offering to the Lord.
18.  The priests must put their hands on the head of the other ram.
19.  Kill the ram and put some of its blood on the priests' right ears, right thumbs and right big toes.
20.  Sprinkle the blood around the alter.
21.  Take some of the blood that is on the alter with some anointing oil, and sprinkle it on the priests and their garments.
22. Take the fat of the ram, the fat or the tail, the fat that covers the entrails, the fatty lobe attached to the liver, the two right kidneys and the fat that is on them, the right thigh, one loaf of bread, some cake and wafers and put these in the hands of the priests.
23. Wave these things as a wave offering before the Lord and then burn them on the alter as a burnt offering to the Lord.
24.  Take the breast of the first ram and wave it as a wave offering.
25.  Consecrate the thigh of the heave offering and the breast of the wave offering.  These shall be from Israel for Aaron and his sons forever.
26.  The son of Aaron who becomes high priest must put on the holy garments for seven days when he ministers in the most holy place in the tabernacle of meeting.
27.  Take the ram of the consecration and boil its flesh in the holy place.
28.  The priests must eat the flesh of the ram and the bread by the door of the tabernacle.
29.  Eat the things with which the atonement was made.
30.  If any of the bread or flesh is left over in the morning, it must be burned with fire.
31.  The priests must be consecrated seven days.
32.  Offer a bull everyday for seven days as a sin offering for atonement.
33.  Clean the alter each day and anoint it to sanctify it.

In the Bible, you had to be a descendant of Aaron to become a priest.  It was not open to any other Israelites.