I don't play a dentist on TV, well, because I am one in real life. Self extraction is a tricky business. Best left to the insane. Now if you have a buddy who needs one, that is different. It is not very difficult to do a simple extraction on a single rooted tooth as long as there is a reasonably stable crown and no crazy curve to the root (that'st why we insist on X-Rays). I think the lay person with just a little training, and the proper instruments, could do it. The problem is being able to recognize when you are in over your head. Break that root tip off down deep, then what? Fracture the jaw, oops! How about causing SBE, or a cavernous sinus thrombosis? Boy I wish I had the right antibiotics. Like all things I guess you have to weigh the risks verses the benefits. Then lest talk about anesthesia, mandibular block (for a lower tooth) is a tricky injection, and getting em pass out drunk is for cowboy movies, so I guess you are going to have to skip it and pull it raw, fun. I guess to start things out you could use a flat bladed screwdriver as an elevator. Please sterilize it. The purpose of the elevator is a Cedar said, to loosen the periodontal ligaments, but also to rock the tooth buccaly and lingualy (palataly in the upper jaw) to expand the alveolar bone. Living bone is plastic and will stretch small amounts if done slowly. The elevator can also be used to literally elevate the tooth out, and I have done this, but it takes a long time. Most providers, once the tooth is properly elevated switch to extraction forceps. There are many different styles for different teeth and taste, not sure what I would use out of my tool box in the garage, maybe a pair of channel locks, but boy would I spend a whole lotta time elevating with the screw driver. The pliers are then gently but firmly used to grip the tooth, you do not want to break it, then the Buccal and lingual rocking is continued until the tooth rolls out to the lingual for a lower and buccal for an upper. Then you must compress the bone, and make sure a clot forms. If not you are going to have to place a mattress suture and or pack the socket. Follow up must be done to deal with post surgical infection, dry socket and the likes. Of course this is if it all goes well, if not, your screwed.
HTH