The level of water in the bowl part of the toilet slowly drains after I flush i.e. I flush, the water drains the after it refills the water level slowly lowers to where there is less then a cup. Is there something I can do myself to fix?

There are a few things that could take your level down. There could be a crack in the bowl letting the water seep out over twenty minutes. That would be my first guess. My second guess is kind of far out but since I've had a couple of calls that lost water in the bowls, I'll throw it up for grabs. The water in your bowl is a trap/seal and connects directly to a pipe called a vent located on your roof. If that location is windy, it has been noted that the wind blowing over the vent pipe causes a suction that syphons the water out the bowl. But that's rare and you will notice the level raising and falling as it happens. I'm going to go with door #1 if only because of the time it takes for the level to go down. The only fix I know is to change out the bowl There could also be a factory defect in the bowl. Sometimes in casting the bowl there is a small bubble in the china. Instead of junking the bowl the manufacture will put a porcelain patch over the hole and send it on through. On occasion the patch will fail allowing the water to seep out of the bowl. If that's the case you would have a full bowl after you flushed and the bowl filled but some time later you would notice a drop in the bowl level. The only other thing might be if you dropped something in the bowl and cracked it. If you're worried about where the water's going, in both cases it drains back down into the sewer and not under the bowl on the floor. To check the bowl must be pulled, filled and put up on saw horses. You can then see if there's any leakage from the bowl.
Good luck, Tom