Is the cleverest or the most clever correct? Is cleverest a word?

Yes, cleverest is a word. The cleverest and the most clever are both acceptable to use. The cleverest is much more common. 

The cleverest/The most clever are examples of Superlative Adjectives. This means that you compare one thing with all the other things in a group.

Here is how you form Superlative adjectives

Rule Superlative Adjective Examples
1 syllable(short words)
the + adjective +est
I am the tallest in the class.
You are the fastest sprinter.
2+ syllables (long words)
the most + adjective
She is the most beautiful woman in the world..
He is the most intelligent in his class.
Words that end in Y
-Y and +iest
You are the luckiest in your family.
This is the happiest day of my life.

According to the table, the most clever is correct according to the rules. There are however some exceptions to these rules including narrow/narrowest and simple/simplest. Both are acceptable when the adjective has two syllables and the stress is on the first part of the word. You can’t have two options when the word has three or more syllables

Learn more about Superlative adjectives in my guide here and find a full list of superlative adjectives here.

The cleverest can sound a little strange to your ear because of the -er at the end of clever. Many people add -er to every comparative adjective when they are learning English(children and non-native speakers).Comparative adjectives compare two things.

Cleverer or more clever?

Cleverer and more clever are both acceptable. These are examples of comparative adjectives.