How can a cohen marry a convert halachically by Chupah and Kidushin? Is this possible?

Shalom, The laws of a cohen are very special, and beyond the special duties he will perform in the Temple (when it is rebuilt), even today he carries out a special function amongst the Jewish people. The blessings of the cohen and the honor accorded to him, still mark him and his lineage as very important for the spiritual welfare of the Jews. Together with this task and honor, come certain obligations and restrictions. The most famous of these is that he cannot come in contact with a dead body. There are also restrictions on whom he may marry. He may not marry a divorcee, a convert, or a chalalah (a women of defective priestly status). Therefore, in answer to your question, there is no way a cohen may marry a convert halachically. If for some reason they did get married (by mistake, or they hid the fact of their status and had a wedding service) they are married after the fact. However they are under constant obligation to get a divorce from this prohibited marriage. In practical terms, it is always worthwhile for a couple in this situation - hopefully before the marriage, but also afterwards - to go and speak personally with a learned rabbi, as in some cases there are details of the priestly status that can affect these laws. For example, sometimes the supposed cohen is not really a cohen, or has halachic disqualifications to his lineage, which then allow him to marry a convert. Blessings.