that particular box is an Arlington product.
stubbie got you the hole in the wall but you need to get the wire up into the box. With a basement, that makes it a bit easier.
Do you know the structure of the wall? Wood framed, block, double or triple layered brick (some old old homes were built in this manner).
So, hopefully it is a wood framed wall. If so, when cutting in for the box, you may need to cut through the plywood sheathing as well (you will need a hole in it anyway). Measure from some reference point that you can use both in the basement and outside. Then, you will need to drill through the bottom plate of the wall directly under where you cut the hole in the brick. You should then be able to use a stiff wire (coat hanger) to poke from the bottom up (or vice versa). use this to pull the wire to the box cutout. Pull it into the box and mount the box to the wall and finish the install.
If the inner wall is CMU or brick, it changes the level of work.]
Let us know what the wall build is.