The problem here is that you're saying 'the key of C' and that doesn't really give anyone the whole picture. Do you mean C major? C minor? C aeolian? C mixolydian?
If you mean the key of C major, you can play notes from the C major scale, CDEFGAB, and then of course you can play all the C major modes so D dorian (DEFGABC) E phrygian (EFGABCD) F lydian (FGABCDE) G mixolydian (GABCDEF) A aeolian (also called the relative (natural) minor) (ABCDEFG) and B locrian (BCDEFGA). Of course you'll have noticed that those are all the same notes in different orders, so it's all largely irrelevant. In the key of C (major) just use CDEFGAB...
Use the C minor scale in the key of C minor.
When people say 'the key of C' they usually mean C major.
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