My office is in the Atlanta area, so I don't typically practice in Chatham County; however, the process should be quite similar to that in our area.
Before filing an eviction, you first need to make a demand for possession of the property, typically in writiing. After doing so, you can file a dispossessory (eviction) in Chatham County Magistrate Court, either on-line (https://www.ncourt.com/nCourtEFile/Default.aspx?Juris=GAChatham) or in-person (http://www.chathamcounty.org/Chatham/Dispossessory_Form.pdf).
Generally, if you are looking only to evict the tenant, the Dispossessory will be served by tack-and-mail. If you are looking to evict the tentant and get a judgment for past-due rent, you will need to tell the Sheriff or Marshal's Service that you need personal service.
Once served, the tenant has 7 days to file their answer. If they don't you are entitled to a Writ of Possession (a court order saying you are entitled to possession of the property). Contact the court on or after the 8th day after service, and they can assist you in any further paperwork. There may also be an addtional filing fee.
The Writ of Possession is then forwarded to the Sheriff or Marshal's Service so that the eviction can be scheduled. Note that landlords generally need to provide their own labor to remove any tenant possessions upon the property....the Sheriff/Marshall merely supervises the process.
If the tenant answers, the Court will set a hearing and all parties will be notified. You will need to attend and be prepared to testify to the Court to support your case. If you win (or if the Tenant doesn't show), you will be entitled to a Writ of Possession.