Just enough oil so it doesn't "fry" or "sautee" but enough so it has the chance to brown and actually cook. The hard part about doing this initially is potatoes soak up a lot of oil. Be sure and start off with a cold skillet. Let it heat heat. Add your butter/oil combination, let that heat up then add your shredded potatoes. Use skilletlicker's method and you should be ok. If you notice the potatoes are not actually cooking, but just browning too much, add a bit of oil by holding up an edge of the hashbrowns and add some butter/oil to the underneath side.
"Count yourself...you ain't so many" - quote from Buck's Daddy