I’m looking for advice with the uchi (hammer-on) technique covered in Shamisen Crash Course: Volume Two. I understand the mechanics of it, I can do it when practicing the technique, and I also do a semi-decent job on measure 1. However, every attempt at measures 2 & 3 results in my muting the string. Adjusting how I hold the shamisen (I may have been tilting it upwards slightly to see the strings, 
) did not help. What else can I try?
I have the same issue but so far no great solution 
I found some information that might help. 
I read in the Kamigata Shamisen book:
[Your fingers] should be pointing mostly vertically from above. Not only that…there should be about enough space left in your palm to fit a ping pong ball.
My left-hand fingers were angled horizontally (which is how I normally play ukulele or guitar), and I wasn’t maintaining The Claw™ which Kyle talks about in his book. (Shamisen of Japan, p117)
I found that when I made corrections to my form I was finally able to do the uchi (hammer-on) while playing the song (or at least the first four measures).
You have to try around with the angle of your fingers, also it depends on stylistics if you want to set soft or powerful accents with Uchi’s.
But in general, for a strong & clean Uchi: With your fingertips, and really really strong. Seriously, think of it like smashing your finger through the string, into the fingerboard.