A kohen is a patrilineal descendant of Aharon HaKohen and must also be “kosher” from his matrilineal ancestry — meaning that his mother, grandmother, great-grandmother etc. where all rabinically fit to marry a their kohen spouses.
In the absence of kohen meyuchas certification, modern kohanim fall into two general categories — religious from birth and baalei teshuvah.
A kohen of the former sort has the status of kohen muchzak - this is so even in the absence of a lineage document.
What is the bottom line answer of who is a kosher kohen? its simple; the kohen who is recognized as such by his kehillah kedosha is deemed a kohen muchzak –a “status-quo” kohen. this is no small title as it carries with it numerous restrictions, responsibilities, and even privileges.
The kohen muchzak is considered a full and kosher kohen מן התורה (Min HaTorah –biblically).
Working in favor of the authenticity of todays kohen is the value of the term kohen muchzak; the entire concept of chazakah in halacha is founded on the base of the kohen being what he and others think he is –an authentic and kosher descendant of Ahron HaKohen. But again, this is just scratching the surface of a number halachic angles (such as all family’s have the chazaka of being kosher, an absence of A valid disproof of the kohen’s sound chazakah) that place the kohen muchzak as a fully authentic, kosher kohen.
The sdei chemed
The sdei chemed, a noted posek is one of many poskim who composed responsa to the issue of kohanim muchzakim –kohanim who are “status-quo” in their kehuna. He is one of the few though who did a thorough work of collecting, analyzing and deciding between those opinions discussed during and prior to his era; The sedei chemed on the kohen muchzak.
Kohen baal teshuva
The kohen baal teshuva often knows for certain that he is of patrilineal descent from a kohen, yet may be unsure of the kashrut status of his matrilineal line. Although we present a general overview of the halachic parameters involved, the halachic status of the said kohen is to be determined on a case-by-case basis by a local orthodox rabbi.