To identify a brown recluse spider you need a very strong microscope and a spider expert.
And an actual brown recluse, which might be harder to find than you think.
These guys are tough to identify even by the experts. Unless you actually have a specimen to analyze, there's no way to know what it was that bit you or that scurried across the kitchen floor. It takes close examination of the spider itself to rule out all the other potential species that look like a brown recluse but don't pack nearly the same punch.
Likewise, you can't tell a brown recluse bite by the wound. There's no blood test or culture that can show presence of brown recluse venom in a suspected spider bite. There's no classically reliable pattern of signs or symptoms to help pinpoint this particular species.
If anything, identifying a brown recluse is more about ruling out what it isn't rather than figuring out what it is. I can't really help an amateur arachnologist identify a brown recluse, but using these steps I can help you figure out the other 99% of the time when it's not.
First, we start with where you found the spider.