I think that your post is more about constructing a raised bed than design so responses may be limited. A raised bed is a good way to grow plants in poor soil. There are a couple of practical things to consider. First is whether or not you should have flowers at your mailbox. Stinging insects may not be appreciated by your mail carrier. Second, grass will get through the cracks in the blocks (if you don't mortar) and into your raised bed if you don't have a border of mulch or hardscape between the grass and the wall. Third, it is difficult to make a mailbox planting area seem like an integral part of your landscape plan.
That said, I would recommend digging the blocks in a few inches to provide a clear and level foundation. There is no need to bury the entire first course for the height of a raised bed. The raised bed should be more than 8" high to leave room for a couple of inches of mulch and give the plants a good root zone. I would also use good top soil rather than potting soil (better water retention) and be sure to remove the grass at the bottom before adding the soil.

Re. burying an entire course: consider what you're going to see. Would it bother you to see 2" of the buried course and an entire course above that? Will the 2" bottom course be concealed by grass (safely distant from the blocks themselves, of course -- as per stevega's caution about grass in the cracks) or mulch? Or will the partial and complete courses merge together?

And consider too how high you'd like the bed to be. And how wide it will be. Draw a sideways view to scale; add the mailbox; add the plants at the top -- what do you think of the proportions? Will you be planting anything that will hang down over the side of the bed? How far will it hang down? Maybe you'll end up burying an entire course, then adding two courses on top of it.

You could also take a look around to see what has been done in the area in the way of raised beds around mailboxes (I would do this mostly to make sure the Postal Service doesn't object to the sort of thing you're planning -- maybe ask the letter carrier or call the local post office). My mailbox is across the street -- that's the side the letter carrier delivers to. If it were my own property, I might be able to build a bed to the sides and the back of the mailbox, but not immediately under the box: that would be demolished by either the letter carrier's vehicle or the snowplows (in fact, my next door neighbor's mailbox was uprooted by a snowplow a few months ago).

What size blocks are you using, how do they go together, and how high is your raised bed going to be above the existing grade?

depending on how tall the end results will be, you may or may not need to bury the one block.

I have two rows high of brick, and have not buried them. I did put a trench of sand under them, though, to keep them from sinking down.

Hope that helps!