Best way to determine if a list of bytes are random? Is there some algorithm out there that can return some value indicating a level of randomness? I believe it's called Data Entropy.

I recently read this article: http://faculty.rhodes.edu/wetzel/random/mainbody.html

Would his approach of analyzing coin flips apply for bytes? Should I drop down to the bit level where it's true/false again or is there a way to determine based on the full byte value?

Are their better analyses than this article?
In TCS, another approach to this problem has been via property testing of distributions, where one is to distinguish whether a distribution is (truly) uniform distribution or is "not even close" to being uniform (in a formal way). Here one gets precise bounds on the number of samples needed to decide on the question.

See, for example Section 6 of the following tutorial:
http://people.csail.mit.edu/ronitt/papers/icm.ps

In particular, one can decide whether a distribution on $[n]$ is truly uniform or is $\epsilon$-far (in total variation distance) from uniform with $O(\sqrt{n}/\epsilon^4)$ queries/samples from the said distribution. (This is also tight in the sense that $\Omega(\sqrt{n})$ samples are needed.)