The closest I have been able to figure out, through google searches, how to write out the phrase "become one with shadow" is Hitotsu Ni Naru Kage (一つになる影). Is this correct? If not, how would one say this?
You're pretty close, but not quite there.
一つになる影 is not
become one with shadow. The reason for this is because if you use the structure
verb (plain form) + noun, the English equivalent is
noun that verbs. Here's an example:
Person/people that run(s) marathons
Therefore, your attempt translates as follows:
Shadow that becomes one.
What you need is a little bit of reordering, and a particle that means
with as well. In Japanese, that particle is と. Simply put the と particle after a noun, and you get the result you seek. Please note this is just one of several uses for the と particle, the other functions you will have to research separately. So here's an example:
Go with him to the store.
Extending these changes to your attempt yields:
Become one with (your) shadow.
Context here matters. If you're talking about someone trying to hide in their shadow, the 'your' will be added in translation even though the Japanese doesn't change. If you're talking about a ninja becoming one with shadow to infiltrate an enemy camp, the Japanese won't change, but the translation will reflect this. As you study the Japanese language, you'll get a feel for what is and is not implied.
Hope it helps.