Little is known about the clinic before the outbreak began, except that it used to be part of the Town Hall.
After Woodbury was secured from Walkers, facilities were being built or re-established to keep the citizens within its walls idle. One of these would be The Woodbury Community Clinic, where the resident, Dr. Stevens, supervises medical operations.
After being found by Merle, Andrea and Michonne were taken to the clinic, as Andrea had a fever. Merle and Michonne stayed in the room, along with Dr. Stevens to check up on Andrea and to talk about Woodbury and her and Merle being 'outsiders' of the group. Andrea stated that she wasn't an outsider, she just got left behind.
The Governor himself was admitted to the Clinic after Michonne stabbed a glass shard through his eye. Dr. Stevens cleaned his wound and placed a bandage over his eye, before he left to address the public of Woodbury about the prison group, and have Daryl and Merle fight to the death.
After the events of "Welcome to the Tombs". The entire place along with the rest of the town is abandoned.
The Governor returns to Woodbury after being abandoned by Martinez and Shumpert. He drives a truck through the main entrance gate and proceeds to burn the town down. Destroying the clinic in the process.