The a in alive is a linguistic fossil:a worn-down form of the preposition on, which in Old English had a variant form an. The form is treated at some length in OED 1, s.v. A, prep.1.
This preposition was still common well into Early Modern English—indeed, it has lasted in dialectal use to the present day. It might of course be used with any noun or noun phrase, and many short phrases which were in frequent use have survived as either adverbs or adjectives or both. In Modern and Present-day English, however, these phrases are now written and understood as single words, the prepositional sense of a having long since been lost. A few you may have run across are aback, abed, afar, afire, afoot, aground, ahead, ajar, aside, asleep and athirst. The a in the middle of nowadays has the same origin.
Note that all the forms which follow a are nouns; and the same is true of the -live in alive, as is clear from the pronunciation with the ‘long’ vowel as opposed to the ‘short’ vowel in the verb live. Alive is thus an ordinary adjective, with no verbal sense, and there is no justification for the verbal form aliving.
But you should not feel bad about this mistake; it’s an old one among native speakers, although not specifically with regard to alive. OED 1 observes of the original noun in the construction that
being often the verbal sb. [substantive] of state or act, it has been in modern times erroneously taken as a verb, and taken as a model for forming such adverbial phrases from any verb, as a-wash, a-blaze, a-bask, [...] These are purely modern and analogical. (ibid., 11.)
It should be added that since this fascicle of OED 1 was published in 1884, these (mis)constructions have gone out of fashion, and the prefix is no longer either ‘modern’ or productive.