How deep your trenches are depends on how far apart you can place them. Allow enough space in between for temporary dirt storage and walking space.
I think those are the Kennebec potatoes (bought as seed potatoes from a source that guarantees them disease-free; don't plant potatoes from the store as they may harbor disease). We also planted Yukon Gold, Red Pontiac, and a few Blue Adirondack that I had left over from my home planting.
Notice that Barbara is planting whole potatoes (which you can see have started sprouting; this is just fine). You can also cut your potatoes into pieces with about 3 buds or eyes each, and expose them to air for at least 24 hours before planting. This is especially fun when planting blue potatoes, as I discovered:I'll show you the progress of our potatoes as they grow. This year we are planning to use row covers to try to hold off the potato beetles and corn borers.
As an extra, here's a photo from my trip a couple of years ago to Peru (the original home of the potato). These are potatoes left on the ground to freeze-dry (August, so winter, and about 10,000 foot altitude). Potatoes used as a staple food in Peru are often stored dried.This is a mix of just a few of the more than 5000 varieties that have been raised in Peru.
Another growing note: potatoes can be grown in containers. Large containers work best: the black plastic compost bins with holes that Montgomery County gives out free are excellent. Put seed potatoes on a few inches of good soil at the bottom, cover, and fill in as the plants grow.