The run should probably be directly on the ground if at all possible, i.e. except in extenuating circumstances. Scratching in the dirt is such a central part of chicken behavior, it'd be a huge shame to deprive them of that, etcetera.
Whether the COOP (the house part) is directly on the ground vs raised is a tradeoff. Dirt floors are easy and cheap to build, and provide some useful thermal mass in a decent-sized coop; on the other hand they require extra predatorproofing, are prone to flooding, and for a reach-in coop can be hard on your back.
To predatorproof an area that is directly on the ground, basically you have to either use a poured concrete slab or the equivalent (good for coop, not good for run unless it's unavoidable or rats are a known huge problem), or bury very heavy gauge fencewire at least 12-18" deep ALL around, or run a 2-4' apron of heavy gauge fencewire on (or just under) the ground, well affixed to the base of the fence and well pinned down at the edge.
Goo dluck, have fun,
Pat