http://www.therightproducts.com/WindowBoxClassic.html

The plants shown in the link are mostly NOT suitable for
your NW exposure, especially the small conifers, which I
can't positively identify from the photos angle - if they
are either very young, dwarf Aberta spruces, or possibly
stone pines, neither will get sufficient sun. (Incidentally, conifers are evergreen perennials, which will quickly outgrow the box, and require all-winter watering to survive.) The trailing purple flowers are petunias, which again may not get sufficient sunlight - in
addition, to keep them lush and full, petunias like to be deadheaded regularly, a tricky task whilst dangling out an
upper story window! The trailing chartreuse vine is a sweet potato vine (producing NO potatoes!) - once more,
limited sunlight could be an issue.

Here's my suggestion: expect to experiment this first
season, but to be on the conservative side, choose annuals
from a good quality nursery that are only marked "part shade to shade" (nothing that says full sun to part shade).
If you REALLY like the trailing petunia and/or the sweet
potato vine, then go ahead and experiment - but with just
one of each in various boxes, to see how they stand up -
if they don't work, just delete the experiment and other
plants in the box will fill in.

If you can find them (sometimes difficult, which is why I
suggest a quality nursery), there are a few trailing coleus
which are really stunning, and might hang down as much as
5 or 6 feet by summer's end (but you can easily shorten them) - it is my #1 choice for hanging pots/boxes in partial shade to shade. If you're familiar with regular coleus, the leaves on the trailing varieties are much smaller; also look for trailing torenia (purples) and even
trailing lobelia (blues). Begonias "might" work there, but
I'm concerned the blazing afternoon sun will scorch the
leaves.

Good luck!

Common Name: spider lily
Zone: 10 to 11
Plant Type: Herbaceous perennial
Family: Commelinaceae
Native Range: None
Height: 0.5 to 0.75 feet
Spread: 1 to 1.5 feet
Bloom Time: Seasonal bloomer
Bloom Color: Pink
Sun: Full sun to part shade
Water: Medium
Maintenance: Low

Google it and see. You can also put in regular spider plants, or do a mix of both. They'll take the conditions you describe. What I do to keep watering in check is put gel crystals into the potting soil. Sold as Soil Moist.

The spider lily flower is negligible; it's the foliage that's the real stunner.

i find that impatiens and coleus do the best. petunias will flower but there won't be many. near the house without air circulation, i found my begonias just rot and the torenias get mildew too easily.

if you like the structure in that picture, you might use the following substitutes:

conifers - use faux boxwoods or faux conifers. from far away they looked real enough that my neighbor commented "those bushes won't survive the winter, you know.." e.g., a pic taken from target.com website

sweet potato vine - use a chartreuse or golden colored coleus , they do really well in the shade, although i've never seen any varieties that trail in the local nurseries...

trailing pink petunias - you can use impatiens although they don't trail and i'm not really a fan. you can also try petunies there and see how many flowers you get. it will still be pretty with only a few flowers and the structure lent by its trailing form. alternatively, try a bright red or pink coleus.
impatiens:

coleus

petunias (i'm partial to the millionbells)

as a last note, if you plant coleus, don't take it out until after memorial day or else it will rot in cold wet weather.

i hope that helps or provides you with some ideas. good luck!

do you overwinter your liriope and heuchera in the garage or do you treat them like annuals?

anyway, a very good idea :)