If a monkey types each letter of alphabet exactly once, what is the probability of the word "Hamlet" appearing? A monkey types each of the 26 letters of the alphabet exactly once (the order is random). What is the probability that the word Hamlet appears somewhere in the string of letters?
Progress
So far I thought that Hamlet is a 6-letter string and could appear in 21 different spots in the 26-letter string. But what should I do next? That is, what is the number of possible sequences of the other letters and the total probability?
 A: HINT: First count how many possible $26$-letter strings there are. Now, abbreviate Hamlet to H, and use the remaining $20$ letters along with H. 
A: Hint: For each of these $21$ places in which you can put the substring hamlet the other letters can be permuted in $20!$ ways, for a total of $21\times20!=21!$ strings containing the word hamlet. How many strings of $26$ distinct letters can be typed?
A: If the 26 letters are the only keys the monkey is allowed to press then my answer is 0% because the monkey wouldn't be able to change from upper case to lower case because that "Shift" key is not one of the 26 allowable keys typed.  hamlet or HAMLET would be possible but not mixed case Hamlet like the question states you are looking for.
As the original question is written, my answer seems correct so if you meant hamlet or HAMLET or the Monkey will always type a capital H and always lower case for all the other 25 letters then please clarify that.
A: Alessandro and Ted have both given good hints.
Some things to keep in mind are the following: 
1.) The monkey is typing each of the 26 letters of the alphabet, not just 26 random letters on the page. Thus, instead of $26^{26}$ possibilities for the message typed, we are left with a much smaller number.
2.) Since the strings we're interested in have Hamlet written somewhere, your intuition about having $21$ locations to place Hamlet is correct. However, in each of these states, the $20$ remaining letters (all unique remember) can be permuted in any fashion. So, if we have $21$ places for our 'Hamlet' block and in each position the remaining $20$ letters can be permuted as desired...
If you can calculate the total number of possible messages from (1) and the total number of messages with Hamlet in them from (2), you've got it.
