I am wondering what the process is to be eligible to marry a kohen. I converted a few years ago and I believe my heritage is Jewish but i have not completed my family history.
The only way would be to get absolute proof that you are really Jewish. You would have to be able to document (with proof that a bais din can accept) that every woman in the female line (mother to daughter) was Jewish with no gaps.
That is, there needs to be absolute proof that each woman in the chain is indeed the daughter of the preceding woman in the chain as well as absolute proof that the first woman in the chain was Jewish.
For example, if one of the mothers did not have a birth record, then she could perhaps have been the child of a first marriage or adopted. Since we no longer have the proof, then we cannot continue down the chain.
If one of the women had married a Jew, divorced (without a get), and then had a child by a different man, that child would be a mamzer. This is because the first marriage (according to Jewish law) would not have been ended and the children of the second "marriage" are children of a married woman by a man who is not her husband. tThis would disqualify all the following women in the chain.
Can a Kohen Marry a Woman with a Non-Jewish Father? also points out that a kohen cannot marry a woman with a nonJewish father, though if he has done so he is not required to divorce her.