I'm assuming you plan to grow vegetables in them.  If so, you want to have close to a neutral pH (7.0) and most soil mixes will be 3-way or 4-way mixes (combination of clay/silt, sand and organic matter.)  However your boxes are really deep and depending on the size of these boxes, purchasing all topsoil can be expensive.  One trick I use in my own garden (I did this just last weekend so it's not too late) is composting straw (not hay) and 40# blood or feather meal over the winter.  My boxes are small (3'X4') and two straw bales fit nicely into them.  I then evenly spread the meal over the top and water in.  The bales are now covered with about 4" of topsoil.  It stands about 30" high (above my 24" boxes) but over the winter it will compost/compress and will be ready to plant  in early spring.

In your case, you don't need to used the whole bale but you can put a layer of 10-12" of straw and then sprinkle with the meal, then fill the remainder with soil.

Another tip is to lay down hardware cloth (19 gauge or heavier, 1/4" grid) at the bottom of your boxes prior to filling.  This prevents unwanted critters from tunneling into the boxes.

If you are go the soil route, you may want to wait until spring to purchase it unless you want to plant crops to over winter (hardneck garlic and shallots are my favorite things to overwinter.)  During the winter rain and snow may compress your soil.  You can protect it with leaves or wood chips but you will want to remove that mulch prior to planting.

You are better off with a semi-permeable covering like leaves, wood chips, even burlap is good (if you can get used burlap bags from a coffee roaster, that's a great, cheap source.)  It's really the compaction you are trying to protect against.  The rains will do some leaching, but in the spring/summer when you are planting, you may want to do a quick pH check and amend based on what you want to plant.  Peas and beans don't really need much amendment, but tomatoes really need a good dose of calcium (lime or bone meal are typical choices.)

I wouldn't recommend plastic over the winter as the likelihood of the water puddling could cause more compaction in places.  However, placing black plastic over your beds a couple of weeks prior to planting can be beneficial in warming up the soil and allowing for moderate drying out.