"Doing alright" sounds like a good answer to "How are you doing?".
Is it an acceptable answer to "How are you?" too?
"Doing alright" is an elided sentence, a shortened form of:
I am doing alright.
When someone asks, "How are you doing?" it may seem natural to repeat the doing:
However, when someone asks, "How are you?" it may seem natural to repeat the "to be" verb:
Since the full version contains both am and doing, the shortened form is fine for either question.
Incidently, most dialogues that begin with "How are you?" or "How are you doing?" are followed up with stock phrase answers that don't really need to answer the question using a parallel construct. It's just cordial small talk. I've heard all of the following answers to my "How ya doin'?" queries, and I've never bristled at the grammar or questioned the appropriateness of any of them:
That last one is not very common (in fact, I've only known one person who used it regularly); however, I've always thought that was a great answer to a ubiquitous question.