If you have a care manager on the case, let him or her handle it. Care managers do the matching and hiring, and if any problems arise, the care manager
should handle them. A good care manger might be able to step in and fix the root issue, for example, before resorting to firing.
If you hired the personal assistant through an agency, the first step is to go back to the agency and ask your aide's supervisor or the owner to resolve the issue or give you a replacement. They should work with you to assess what worked and what didn't work, so that the same problems don't arise with the next caregiver.
If you hired the personal assistant independently, you'll need to do the firing yourself. Just talk to the assistant, explaining your reasons and promising to abide by any contract or agreement you set up ahead of time. (I always advise those hiring independently to set up a contract that specifies what happens if the caregiving arrangement doesn't work out. Be clear up front and have your terms in writing: You might agree to give at least a week's notice and a week's severance pay, for example, or two weeks' notice and two weeks' severance.)
When things go bad, if you don't have a legal agreement in place, it can get really ugly. For example, I've heard of caregivers trying to get worker's comp, saying they got injured on the job. If you don't have an agreement in place, talk to the personal assistant and try to agree on an arrangement that seems fair, such as a week or two of severance pay, depending on how long the assistant worked for you.