They used to say it was important to get the whole tick out, but that is no longer the case, the main thing is to remove the body part as soon as possible, preferably within 24 hours, to reduce the risk of being infected. You can use tweezers or your nails, try to grab it as close to the skin as possible and pull it out staright. Don’t use oils, Vaseline, butter or similar stuff, this will just make it more difficult to remove the tick, but clean the skin with soap water or a disinfectant afterwards.
You are right, having deer nearby increases the risk of being bitten by ticks.