-using your C chord (I take it you mean C major) start with the first note, so in this case the root note. You'll be fretting with your third finger, at the third fret, 5th string (A string).
Fretting just that one note, play all five strings starting with the A string, picking them one at at time. Make sure they sound perfect. Don't strum, don't worry about playing a rhythm, just make sure each string sounds clearly. When you've got that...
-add the next finger, your middle finger... 4th string (D string) second fret. Same as before... make sure each string plays cleanly, keeping both fingers fretting their notes. If adding the middle finger causes a string to buzz, identify which string it is, and see if you can get fix the problem. It really should only be the open G string, because you are fretting above on the A string (your fretting at the third fret with your third finger, if you happen to touch the A string with your middle finger as you fret the D string at the second fret, it shouldn't make any difference in the sound of the C note being fretted on the A string). Make sure you understand all that...
-finally, add your first finger on the first fret of the B string. Again, pick slowly through the chord and check for any problems.
If you do have problems as you add fingers, its really likely that its not because your fingers are too "fat" but because you are twisting/moving your fingers as you try to fret with them all at the same time. That's why you start one note at a time, and resolve the problems as they come up.
Really, for a C chord you have plenty of space because as I said above, the only potential problem string is the G string. If you're too close to the G string, feel free to move your finger away from it so its closer to the A, since touching that string won't be an issue.
I've played guitar for a long time, and yet still learn new chords the same way. Whenever you look at a chord shape, its good to recognize where the potential problem spots are, and where you have a bit more "room" as it were.
