Green Acres,
I did a very similar tower for 2 years. Last 2 due to some family issues we were unable to get a garden in in time. I'm going to try it again this year but I'm changing it as my results were mediocre at best. I used 2x4 for the corners and 5/4 deck boards for the side and it was about 2'x2' square and about 3' tall and that part all worked great. The plants grew great and every week I kept burying about 2/3 of the plant till it was near the top. Come harvest I got like maybe 1.5lbs of taters. I videoed it but think I never did finish the edit and post to Youtube as I was so disappointed.
Now here is what I've learned on this as I'm hell bent to get better results. The box needs to be bigger. It means more dirt but part of the problem is that the tower dries out so quickly even with daily watering and the soil also gets very hot. I'm gonna go to a 3'x3' this year and I need to place it where maybe it gets more afternoon shade. Your soil mix needs to be very loose and say about 1/3 peat moss.
Another huge thing is the type of potato. This seems to be the elusive part and is all conjecture to this point for me but potatoes are apparently like tomatoes in that there are determinant and indeterminate varieties. We need the indeterminates and these apparently tend to be reds and late season varieties. I know Yukons didn't do it. The few potatoes I had were at the bottom of the box at the seed level and not up higher in the tower. I'm gonna try to find norlands, chieftains
From what I experienced and the research I've done I plan to put in about 4-5" soil and mix in a little bone meal, plant the potatoes and cover with only 5-6" more and top dress that with a decent coating of more bone meal and water in everything very well. Once they pop thru let them get up about 8-10" and bury 2/3 and keep repeating till I get to about 2 inch from the top. If you fill to the very top it makes trying to water a pain as it wants to run off. I'm hoping to try 3 different boxes. 2 with different varieties of reds and in one I might try yukons again but what I'm thinking is that every time I get ready to "hill" up I might put a few more seeds in between the plant stalks already growing. That way if they are only setting tubers at the level of the seed maybe putting seeds on more layers will let them fill in the "floors" of the tower. Again, sort of a science experiment but I do plan to use the bone meal in all them as it's to really help root crops.
I might try a 4th experiment of just going out to my in-laws who live out on the mountain and raking up a load of leaves from the woods, with some of the mychro fungi in the leaf litter and use that as the "soil" to grow in. Maybe I'll mix in some old or rotted straw if I can find it. I don't know the method of gardening but I saw where a couple just rolled out old straw on the ground and planted taters in it and they did very well other than trying to be sure to keep the tubers covered from turning green. They had great results and the potatoes were super clean. Sorry for the long reply but hopefully you will find something useful here.
Check this guys vids out as he really goes into the biology of the potatoes he grows in buckets. Here is a good one he goes into detail on how potatoes grow.