Every fingering that fits a hand
x101
open · no 5
6464
hard · no 5
6867
moderate · barre
646x
easy · no 5
6467
hard · no 5
What is true of it
C♯m6 is C♯, E, G♯ and A♯ — 1, b3, 5 and 6 counted from C♯.
These are also the notes of A♯m7♭5/C♯ and B♭m7♭5/C♯. Which name is right depends on what is underneath and what came before — the notes alone do not decide.
On an instrument that cannot reach all 4, the 5 goes first, in that order.
The easiest of the 5 fingerings below is x101: 2 fingers, held at the 1st fret.
1 of them use open strings, which is what makes a shape quiet to change to.
1 of the 5 need a barre.
It leaves out the 5. 4 strings cannot carry 4 notes, and the 5 is the one this chord can lose without changing what it is.
1 string stays silent.
Where its notes are on the neck
The brass dots are C♯, the root. Everything else is a chord tone with its degree printed in it.
With a different note underneath
C♯m6/E
x464
C♯m6/G♯
1101
C♯m6/A♯
3404
An inversion is the same chord with a different note lowest. The name changes to a slash name because the bass is what a listener hears as the foundation.