What is the best time of year to plant a weeping willow?

Cuttings of mature wood of the current year's growth, November to February in a sheltered outdoor bed or planted straight into their permanent position and given a good weed-suppressing mulch. Very easy. Plant into their permanent positions in the autumn. Cuttings of half-ripe wood, June to August in a frame. Very easy.

Be aware that it can be an 'awkward' tree.
Its roots can damage sewers and drains and depending on the species you choose it could grow to 50 feet.
Search the web for details on Salix (willow).

That depends upon your location. Where I live (Northeastern, USDA Zone 6) it is best to plant hardy deciduous trees in the early autumn. (Now!) However, depending on your location and climate, that timing could range from now through early wintertime. I agree with the prior post that the tree can be a threat to drains or septic systems. It is a water-loving tree that will be opportunistic when it comes to finding the water it wants. In my area, a Weeping Willow can grow to over 75 feet with a spread of up to 50 feet. I have one in my back yard that we planted five years ago. When we put it in, it was as thick as my thumb and about five feet tall. Now it is 8 inches in diameter and is 25 feet tall. Of course, I assume that some of the explosive growth is due to me watering it with the waste from my Koi-pond filter!

In early autumn or very late summer.

Now is the perfect time. You want to give the roots time to develope before the ground freezes in winter and root developement stops.
Sring will work, but there is a problem if you have an "early summer" that leads into a drought.

Keep it well watered the first two weeks.