Anyone who is "Muhzak" (presumed) to be a Kohen is qualified to serve as the Kohen for the Misva of Pidyon Haben, as one may rely on this presumed status for fulfilling this Misva. Some have the practice of giving money many times, every time they see a Kohen, in order to ensure that they have indeed paid the money to a Kohen as the Misva requires. This practice, however, is a "Humra" (stringency), and not Halachically required, because, as mentioned, the Misva can be fulfilled through anyone with a presumed status as Kohen.
It is preferable to perform the Pidyon Haben with a Kohen who is a Torah scholar. However, one should not delay the Pidyon Haben to wait for a Kohen who is a Torah scholar to arrive. It is preferable to perform the Misva in its proper time with an ordinary Kohen then to wait for a Kohen who is a Torah scholar. Furthermore, if one has a relative who is a Kohen, he may be given preference over a Torah scholar, even if he is ignorant in Torah, in the interest of avoiding strife and hard feelings.
If possible, one should perform the Pidyon Haben with a Kohen who is poor, as one thereby fulfills both the Misva of Pidyon Haben and the Misva of charity.
A person who was born from a union between a Kohen and a woman forbidden for Kohanim such as a divorcee - does not have the Halachic status of a Kohen. Therefore, he is not qualified to receive the money for a Pidyon Haben. Furthermore, one should not perform the Misva with a Kohen who publicly desecrates Shabbat, as this undermines his status of Kehuna to some extent, as indicated by the disqualification of a flagrant Shabbat violator from Birkat Kohanim. A Kohen who worships idols may not perform Birkat Kohanim, and public Shabbat desecration is akin to idolatry. By the same token, a Kohen who flagrantly violates Shabbat is disqualified for Pidyon Haben.
Likewise, one should not perform Pidyon Haben with a Kohen who publicly eats non-kosher food. A Kohen who publicly ignores the special laws of Kohanim such as by marrying a divorcee or going to cemeteries is also disqualified for Pidyon Haben until he performs Teshuba.
Summary: One may use for Pidyon Haben anyone presumed to be a Kohen. It is preferable to use a Kohen who is a Torah scholar, but not if this requires delaying the Misva or insulting a family member who is a Kohen. One should not use a Kohen who publicly violates Shabbat or the special laws of Kohanim, or who publicly eats non-kosher food.
