If you use the C chord shape that you know with D6 tuning (aDF#B), that would actually be a D chord. All chords would be 1 step up.
Code:
C6 D6
--------------
A -> B
A#/Bb -> C
B -> C#/Db
C -> D
C#/Db -> D#/Eb
D -> E
D#/Eb -> F
E -> F#/Gb
F -> G
F#/Gb -> G#/Ab
G -> A
G#/Ab -> A#/Bb
If you're playing alone, you could use the same chord shapes but you would be playing in a different key; a song in the key of G in C6 (gCEA) would be in the key of A in D6 (aDF#B).
If you were playing along with someone who is using C6 tuning, you would need to use different chord shapes.
I'm really glad someone addressed this here, I love the way the D6 tuning sounds, but I didn't know anything about it at all. I can be such a dunderhead. Thanks!