First off, your Dictionary is wrong: you want a count of repeating characters, so your Key needs to be the unique part - the character - and the value should be the number of them
Dictionary<char, int> charCounts = new Dictionary<char, int>();
Now you can check if a character exists in the dictionary each time you come across it, and either add it or increment the count.
Secondly, your code doesn't check for letters at all, it just loops on the strings in the array, and copies them into a dictionary, using the array index as the count - which is completely pointless.
Go back to your homework question, and read it carefully. If you are supposed to count characters, then that means processing each string in the array, not counting strings at all.
I suspect you are supposed to find either "how many 'H's are there in the array of strings?" or "how many strings have an 'H' as the first letter?" - but either way you need to think a little more carefully before you jump into code. Remember, a dictionary cannot have duplicate Keys ...