This is one of my favorite solo pieces for bass. It has modern harmony, rhythmic variety, it lays very well on the instument, it's lyrical and has some flash. What's not to like? First, get the recording of Frank Proto playing this piece. It really gives you some insight as to what he was thinking. A few things off the top of my head...
Mvt. 1 - Don't go too overboard with vibrato. The line has a very natural arch to it through the movement, and I believe there should be a slow and steady build to a peak which then resolves all the way to the low D.
Mvt. 2 - This may be the hardest movment for "classical" bassists. Put the bow down. Swing it hard. I think the indicated tempo is a bit fast for what is happening in the music, so don't be afraid to back it off. Personally I really like this movement at around 130 bpm because it can get into a nice groove.
Mvt. 3 - Romantic and mysterious. Really try to let the artificial harmonics sing.
Mvt 4 - Burn it down. The ends of the 3rd and 4th movements are the same, and there are some logical places to shift if you use the harmonics as guide points. Make the 16th runs as crisp as you can using a very controlled bow. You don't want to saw a tree down at 160bpm.
I found the trickiest part to this piece is finding a pianist that can play it, and then coordinating the music with them. The piano part is very dense and reminds me of something Hindemith might write. I did an analysis of this piece for a class during my undergrad, and I'd be happy to talk about it further if you'd like.
All the best,
Pat