The certainty - Have full evidence or proof of.
Been instructed - By the preachers of the gospel. The original word is the one from which is derived our word "catechism - been catechized;" but it does not denote here the "manner" in which the instruction was imparted, as it does with us, but simply the fact that he had been taught those things.

Wherein thou hast been instructed - Κατηχηθης - In which thou hast been catechized. It appears that Theophilus had already received the first elements of the Christian doctrine, but had not as yet been completely grounded in them. That he might know the certainty of the things in which he had been thus catechized, by having all the facts and their proofs brought before him in order, the evangelist sent him this faithful and Divinely inspired narrative. Those who content themselves with that knowledge of the doctrines of Christ which they receive from catechisms and schoolmasters, however important these elementary instructions may be, are never likely to arrive at such a knowledge of the truth as will make them wise unto salvation, or fortify them against the attacks of infidelity and irreligion. Every man should labor to acquire the most correct knowledge, and indubitable certainty, of those doctrines on which he stakes his eternal salvation. Some suppose that St. Luke refers here to the imperfect instruction which Theophilus had received from the defective Gospels to which he refers in Luke 1:1.

That thou mightest (e) know the certainty of those things, wherein thou hast been instructed.
(e) Have fuller knowledge of those things which you know only partially.

That thou mightest know the certainty,.... The end the evangelist had in writing this Gospel, and sending it to Theophilus, was, that he might be more strongly assured of and more firmly established in the truths of the Gospel. The Vulgate Latin, Syriac, and Arabic versions render it, "that thou mightest know the truth"; that is, the certain truth of things: the truth he did in some measure know before, but Luke's view was, that he might have a more certain knowledge of it; both truth, and the certainty of it may be intended: so the Hebrew word, signifies both truth and firmness; and the word here used signifies such a certain evidence of things, as may be safely depended on; even
of those things, wherein thou hast been instructed; or catechised, signifying, that he had been hitherto taught, as a catechumen, the rudiments, and first principles of the Christian religion, by word of mouth; and he had taken them in upon the evidence they came with, and the authority of those that instructed him in them; and now he sent him in writing this account, to increase his knowledge, strengthen his faith, and to give him such a sure proof of things, as might preserve him safe in the belief of them, from all doubting and defection. Having finished his preface, he proceeds to the narrative itself, which begins as follows.

that thou mightest know--"know thoroughly."
hast been instructed--orally instructed--literally, "catechized" or "catechetically taught," at first as a catechumen or candidate for Christian baptism.