Would my tenant be liable if they "fixed" my outside outlet solely for the car and when I plugged it in today and the car went on fire?

They might be liable. Certainly, if they had "fixed" the outlet without permission, they would be liable for all consequences growing out of their unlawful actions. Even if they had permission, if the fire was caused due to their negligence, or unreasonable carelessness, in how they did the work--e.g. they did the work wrong, or were not qualified to do it--they would likely be liable if they had represented they were qualified to do the work and were doing it in exchange for some payment, or a reduction in or credit against rent, etc.

They would likely not be liable if you asked them to do the work as a favor to you, they were not being paid, and they did not represent any special skill or knowledge: in that situation, it could be the case that you would be found to have been liable yourself to let untrained persons to electrical work; and also, their duty to use care in that case is more limited to begin with, because they are doing the work simply as a favor, and not because they were "hired" in one way or another to do it.

Note that the above discussion presupposes that the outlet caused the fire--to establish liabiltiy, besides a duty to use care and some wrongful behavior (e.g. negligence), you also have to show causation--that the work on the outlet caused the fire.