What kind of a formula do you want?
If we write the formula you posit for $[D]$ doesn't make sense: if we restrict $[D]$ to $V$, we get $[D]_V$; that's a tautology.
The reason that the adjunction formula has a more complicated shape is precisely because the restriction of the canonical bundle of $X$ to $V$ is not the canonical bundle of $V$.
The Chern class of $[D]_V$ is precisely the intersection of (a generic representative of the linear equivalence class of) the Chern class of $[D]$ with $V$. (If $s$ is a generic section of $D$, assuming it admits one, then the Chern class of $[D]$ is precisely the zero locus of $s$. If we restrict $[D]$ to $V$,
then $s_{|V}$ will be a section of $[D]_V$, and its zero locus will be the intersection of the zero locus of $s$ with $V$.)
Based on your question, it seems that you are confused about something at a more basic level. Perhaps asking a more specific question would help.