Can I replace a toilet bowl myself?
Yes you can. 

You will just need to remove the bolts that are at the bottom of the bowl and then raise the toilet up.  I recommend flushing it first as it'll be lighter. 
ALso have a trash bag handy. Although you have flushed the toilet, there's usually still some water in it. I recommend one person lift the toilet up and another person put the bag under the bowl to catch any water that may run out.

Then you'll need to scrap the wax ring off of the pipe in the floor. 

Then install a new wax ring in the same location as the previous one. Do not press on it. 

Then install your new toilet over the bolts and tighten them.