Jim, I would like to point out that your question is an unreasonable one. Asking someone to list chords or to conceptualize a chart that would contain all of the chords a given note ("C" in this instance) could belong in is like asking, "I like Oregano, can someone tell me all the foods I can put Oregano in?"
The truth is, there are an infinite number of foods than can have Oregano in them. Cooking and recipe books do not function in this way. If they did, they would be thousands of pages thick.
That said, you also ask a much more reasonable question, which more acutely illuminates what I believe to be your real question, which is "How do I know what chords to put with a melody?"
The answer is to learn about chord progressions.
In order to learn about common chord progressions, you need to understand a little bit about keys and how they function. In order to explain this without writing a book, I am going to circumvent a large volume of information.
The most common chord progressions heard in pop music involve three chords based off of the first, fourth, and fifth scale degrees of a given key. Let's use "C" as an example.
The notes in C major are C, D, E, F, G, A, B before repeating.
Therefore the most common chords in the key of C would be C (I), F (IV), and G (V).
By creating triads off of each of these pitches, we can see if any of the chords share common notes.
C (I) = C, E, G
F (IV)= F, A, C
G (V) = G, B, D
So you see that C and F both share the pitch "C". This is called a common tone. If your melody has a C and you are in the key of C using the three chords I outlined above, you could harmonize your melody with either C (I) or F (IV) since they share a note.
I would highly recommend picking up a book on beginning music theory. I would be happy to suggest one, but I am mostly aware of textbooks.
That said, you may want to look at "The Complete Idiot's Guide to Music Composition". It was a book I picked up many years ago when I had this same question, and it helped to point me in the right direction.
By all means you should use this answer as a starting point and not an ending point.
Hope this helps.