Once you win and pay the balance, the clerk still has to go through their process before you officially get title. First they’ll issue a Certificate of Sale, then there’s a 10-day window where objections can be filed. If nothing gets contested, the clerk will then issue and record the Certificate of Title in your name. In most counties I’ve seen, you’re looking at around 2–3 weeks from auction day before the deed is in your name, sometimes faster if the clerk is efficient.
As for the former owner still living there, you can’t start eviction until the Certificate of Title is recorded. At that point you file for a writ of possession. Once the judge signs it, the sheriff posts notice and gives them 24 hours to leave. Realistically, that process usually takes anywhere from a couple of weeks to maybe a month, depending on how backed up your county sheriff is.
So in short: expect about 2–3 weeks to get title, then another 2–4 weeks to actually get possession if the owner doesn’t leave voluntarily.
You could also offer the tenant cash for keys, or other arrangements to speed things along and keep it more civil - read If I Win A Bid On A Property With Tenants How Do I Get Them Out?