How can I raise grade near my fence without making soil contact?
Anything directly against the fence will allow moisture to accumulate and hasten decay. If you want a fast, simple solution, lay any rigid board against the fence to the new grade level with spacer shims to keep contact minimal. It'll do the job. Don't use plastic. It'll look bad and trap even more moisture.
Ideally you'd create a retaining wall a few inches away from the fence. Put down landscape fabric and your choice of covering or mulch (washed or crushed rock, cypress bark, etc.) between the fence and wall. It'll look nice and won't create too much of a safety hazard.
You could also do a double retaining wall with a terrace between for shrubbery and/or flowers. This would reduce the appearance of a channel against the fence and improve safety somewhat.
Whatever you decide, I'd mock up a short section of it to get a feel for how it transforms your yard. You might find that what's in your imagination doesn't match reality. Better to know that before you do the entire job.