You can use "more clever" or "cleverer", as the comparative of clever; and you can use "most clever" or "cleverest," as the superlative of clever. However, it is safer to use "more" in most cases when dealing with anything other than a one syllable root word as I will explain below.
In English the decision on when to use -er/more, and -est/most is based on two things: the number of syllables in the root word, and the ending of the root word.
The general rule is, as other members have pointed out, if the root word is one syllable you use the suffix -er, if the root word is three syllables or longer you always use more and not the -er suffix, but in the case of two syllable words, like clever, you can use either depending on the ending of the root word.
Two syllable words which end in: -less, -ful, -ing, -ous, -ant, -ed are never modified with the -er suffix and must to use "more". For example, "useful" cannot be usefuler, something is "more useful"; or someone can be "more pleasant" but not "pleasanter"; or "more boring" not "boringer", and so on.
While the two rules above (syllables and root ending) are hard and fast rules, what I will explain below is a matter of taste, preference, or practice.
In the case of clever, its ending of -er, does not require the use of more, but there was a period of time, at least it the U.S., where the -er ending, the -le ending, and the -n ending were all included in the above, and for that reason "more clever" probably has a more natural sound than "cleverer" to most people. Examples of this for the other endings are "simpler/more simple", "more fun/funner". Since this was taught for some time, using the -er suffix fell out of practice for words with these endings, which means that while correct under either -er/more option, their unusual use seems to make people uneasy, as though it were something a child would say. It is not technically incorrect to say "cleverer", just odd sounding to many people.
Also, since another use of -er in English, is to convert a verb into a noun of someone performing the verb action, for example the verb: to speak can become the noun: the speaker simply by adding -er, it is helpful to avoid applying -er when the word could be confused as this type of -er noun, rather than an -er comparative adjective. For example, even though the adjective common (two syllables without a special root ending) can be modified as "commoner," you would want to make sure that your context is clear so that "commoner" does not sound like you are referring to the noun meaning an ordinary person. When in doubt, or to be clear, you could just say "more common."
Furthermore, when a comparative is combined with another modifying phase, and the root word is two syllables or longer, it is usually preferred to use "more" or "most', rather than "-er" or "-est". For example, if I wanted to emphasize just how clever someone was in relation to someone else by using "a lot" or "much" it is less common to use the "-er" suffix. Example: "Sally is a lot more clever than I originally thought." sounds better than "Sally is a lot cleverer than I originally thought". Or, "Dan is much more clever than Steve." is generally preferred over, "Dan is much cleverer than Steve." This does not apply to one syllable root words, however, which do use the -er suffix even when accompanied with an modifying phrase. If you want an example, replace strong/stronger into the above example; native speakers would prefer to use "stronger."
So my long explanation boils down to this: you can either either, but since there are so many circumstances, due to endings, or modifying phrases, that can cause a two-syllable word to require "more" instead of "-er", I generally default to using "more" which is a perfectly correct option for anything but a one syllable root word adjective.