In general, in a civil forfeiture case, a law enforcement agency seizes the property based upon probable cause that it is connected with a crime and/or is contraband, and then (depending on the nature of the property, the the civil forfeiture statute invoked) gives notice by mail and/or publication to potential owners of the property (sometimes in connection with a civil lawsuit, sometimes in an administrative process).
The notice requirements are much more lax than in a typical civil lawsuit and the time limits are generally quite short (on the order of weeks rather than months or years). If there is no timely response, the property is forfeited.
If there is a timely response, the agency may relinquish the property if they agree they were in the wrong, but otherwise (and usually), the process must be escalated by a claimant into a hearing typically with a preponderance of the evidence standard of proof, at which the claimant must prove that the property is not subject to forfeiture.
The 5th Amendment cannot be invoked by someone seeking to recover property in a civil forfeiture, so making a claim risks providing evidence for a criminal case against the claimant.
The court with jurisdiction is typically one where the property was seized which is difficult for property that was in transit or seized from someone who was traveling.
The rights of non-owners (e.g. mortgage holders) in property that is seized to notice, and to relief if they are not given notice and the property is seized, varies.
It is hard to be specific, because so many different laws are involved. For example, I have the most experience representing people when the postal service seizes property, but have seen it play out in other contexts, each of which has procedural nuances. Indeed, figuring out which civil forfeiture law applies is often the first problem facing a victim of civil forfeiture.