Conviction here means a strongly held belief or opinion, i.e., something one is convinced of. And you'll have to widen the context a bit to find a stated opinion.
But it is a still greater evil to me that I am self-educated: for the first fourteen years of my life I ran wild on a common, and read
nothing but our uncle Thomas's books of voyages. At that age I became
acquainted with the celebrated poets of our own country; but it was
only when it had ceased to be in my power to derive its most important
benefits from such a conviction that I perceived the necessity of
becoming acquainted with more languages than that of my native
country. Now I am twenty-eight, and am in reality more illiterate than many schoolboys of fifteen.
So the letter writer is convinced that the greater lack in his life is that he has no formal education, one that might have instructed him in languages other than his native tongue. But, alas, he has come to that realization too late for it to be of benefit: he's too old at age 28 for such study.