I take it by your posting that you are actually the landlord in this situation even though you are also the person's roommate. The process to evict is fairly simple, but you have to proceed under FL Statutes, Chapter 83. I have posted a guide on evicting a person in FL that you should take a look at.
I am also presuming you will be filing as an individual and not under a company name. If you own the property under a company name, then you can still file without using an attorney, but the court will evenutally require you to hire an attorney if the eviction becomes contested (i.e., tenant files an answer and/or deposits monies into the court registry).
A 3-Day Notice would need to be served on the tenant/roommate first. Again, look at my guide on using the proper language and the date for demanding rent. This step is VERY important. If you mess up on the notice, the court will throw the eviction out and make you start all over again.
As to the time period to re-gain possession, it should only take about 4 weeks. However, since you state you are this person's roommate, it is more likely the person will just leave once they find out the eviction was filed or once a sheriff serves notice on them. Either way, it won't be a happy environment while you go through the process. It might be better for you to stay somewhere else until they are gone.
For proving that the tenant hasn't paid, I am hoping you kept a rent ledger or at least can show bank statements where the rental funds were deposited. Your bank statements should show you a copy of the check deposited which proves the months paid and which months not paid. If you allege that payment hasn't been made for certain months, just plead it in the complaint and the tenant will have the burden to prove that payments were made.