"Deliberate recluse" or "deliberately reclusive"
Solution 1:
That depends.
If recluse is an adjective, then it would be
deliberately (adverb) recluse (adjective, rare) = same as = deliberately reclusive
If recluse is a noun, then it would be:
deliberate (adjective) recluse (noun)
Note that recluse may be both, depending on context, although recluse as an adjective is very rarely used today. In your context I'd say it's a noun, therefore the original sentence is correct.
Solution 2:
There is little functional difference between the two. In technical terms, "deliberate recluse" refers to you, the person, while "deliberately reclusive" refers to your actions. However, in hearing or reading this, someone would get the general idea that you refrain from public activity with intent. Only by nitpicking or over-analysis would the difference surface. Use whichever you prefer.
Solution 3:
The original is correct: "Aspiring autodidact, deliberate recluse."
Let's parse this. "Aspiring" is an adjective to describe the noun "autodidact," which is a type of person. "Deliberate" is an adjective to describe the noun "recluse," which is also a type of person.
This original expression is a grammatically correct parallel structure.
If you want to use "reclusive," the adjective, then you would also need to modify the first part of the expression. For example, you could write "aspiringly autodidactical, deliberately reclusive." But this sounds overly complex and, frankly, silly.
Solution 4:
"Deliberate Recluse" is a noun, whereas "deliberately reclusive" is an adjective. They are both correct. Let's look at the context to see which is better here.
Aspiring autodidact, deliberate recluse.
Since this is a list of descriptions, they should be consistent. "Aspiring autodidact" is a noun, so "deliberate recluse" should be used for consistency.
Expanded to a full sentence, this would be
I am an aspiring autodidact and a deliberate recluse.
vs
I am an aspiring autodidact and a deliberately reclusive.
which, while correct, is less consistent. However, since it is a short list, it is not very noticeable. In a longer list, the different element would stick out. If you wanted to emphasize a particular element, though, that would be an effective way to do it.