The only elements we have about Charles' literacy can be found in this extract of Einhard's Vita Karoli:

He also tried to write, and used to keep tablets and blanks in bed under his pillow, that at leisure hours he might accustom his hand to form the letters; however, as he did not begin his efforts in due season, but late in life, they met with ill success.

Did it affect his rule? Well, Charlemagne implemented, or at least tried to implement, an impressive series of cultural reforms that emphasised the importance of literacy and literary culture, on a hitherto unknown scale in the Early Medieval West (we can only try to guess what were the results of these efforts, however). I am not an expert of this particular strand of Carolingian history, but illustrations of this concern abound, among which the Admonitio generalis (a legislative text of 789, whose seventy-second chapter ordered the creation of ecclesiastical schools), the Litteris de colendis (a circular that further defines some of the objectives of the policy), and several anecdotes reported by Einhard, among which the idea that Charles “subsidised” the compilation of ancient Germanic songs and legends, which had been oral up to this point. But I do not think that we should understand these attempts as a will of Charles only, though he certainly had a great interest in the domain; rather, both Charles' personal efforts to acquire literacy and his legislation to make it more common should be understood as two manifestations of a clerical ideology that had become closely associated with the ruling circles.

/e “literacy” is used in this comment as meaning “the ability to write.” However, it should be remembered that this ability to write was not mechanically linked with the ability to read in many pre-modern societies, simply because not many people needed to be able to write. As far as I know, no-one clearly states that Charlemagne could read, but the fact that Einhard does not mention it specifically suggests, at least to me, that it went without saying for him.