Would it work to use chain link fencing to do the sides and top of the run and then use hardware cloth three feet up the sides? That way the fencing is strong and there is cloth to keep raccoons from grabbing chickens through the holes.
IF your chainlink is heavy gauge (a lot of what's sold for cheap these days is too thin, it does not even reliably retain a determined dog) AND is installed very very correctly (tightly-stretched without wibbliness, and very well connected all around the edges, preferably with bolted connectors rather than clips at the ends) then what you describe is a good setup IMHO for the sides of the run.
I have more misgivings about using chainlink for a run *top*. If it is a high quality premade kennel panel designed to be on the top, then maybe. But if it is DIY it is very very unlikely IMHO that you will get it done well enough that an animal can't simply pry its way thru (if there is any slack, they can just pull two adjacent wires apart and enlarge it into a really quite large hole). Also chainlink is pretty heavy and saggy for a top. I would suggest instead using welded wire of whatever size holes you feel is acceptable, and supporting it well with some crosswise lumber.
And for the bottom, we would bury either hardware cloth or chicken wire. Does it matter if it's buried?
It matters whether you have some kind of digproofing, yes; but it is NOT required that you necessarily bury anything.
Indeed, a horizontal apron of strong welded wire of whatever sort, 2-3+ ft wide, laid just outside the run fencing and securely affixed to the foot of the run fencing, then pegged or weighed down or covered with mulch/gravel/turf/pavers/rubble/whatever, works pretty much as well as burial does, and for a LOT less work
If you do bury the foot of the run fence instead, you need to go down 18" to have reasonably-good assurance that nothing will go under. That is a lot of digging. Personally I recommend the apron approach for nearly all circumstances
Good luck, have fun,
Pat