I am terrified to introduce peanut butter to my 5 year old's diet. He has no food other allergies but he does have asthma and very sensitive skin. Is there any safe, suggested method for introducing foods? Can we get allergy tests first?

Yes. This issue is still being researched. If a child has other allergic conditions like asthma or eczema or if he has siblings with food allergy it is recommended in infants they be tested for peanut allergy and then peanut introduced at young age with thought that this will prevent allergy in future. However in a 5 year old there are no good recommendations for intro. See allergist for test and intro.
No one... ...is born with peanut allergy, but it can develop fairly quickly in a susceptible individual, so an allergy test before introduction would be nearly useless.