What is the correct lewis structure of SCH⁻?
This model seems reasonable, as you can see from the formal charges carbon has the -1 value while the rest are 0. One thing that you have forgot is to add brackets around the structure to show that it is a ion, include its overall charge.

$$\ce{[\overset{..}{\underset{..}{S}}=\overset{..}{C}-H]-}$$

Although, I suppose a structure of $\ce{[\overset{..}{C}#\overset{..}{S}-H]-}$ would work with the same formal charge, except that sulfur has the expanded octet, which is possible but I am not sure if that is more reasonable or not, it probably is not.

It is a very odd ion, to say the least, probably very unstable. Perhaps you have mistaken the hydrogen with nitrogen? It would be more reasonable to be asked to draw the lewis structure for $\ce{[SCN]-}$, thiocyanate ion. I am not even know how to pronounce or name the $\ce{[SCH]-}$ion you are required to draw, or if it even exists. Maybe that is why you need to draw it though.