If you have an index and have measured its beta with respect to the overall market, how would you go about adjusting it against spread dv01 and why would you want this number?

If you believe that the fundamental economic relationship is

$$r_{\text{Spread}} = \beta \, r_{\text{Market}} + \text{const}$$

Then in order to obtain the beta of a credit index $I$ with CD01 $c$ to the market you would write

$$r_I = c \, r_{\text{Spread}}$$

and thus

$$r_I = c\, \beta \, r_{\text{Market}} + \text{const}$$

Now you need to estimate $\beta$ from observed data. To do so, you need a time series of $r_{\text{Market}}$ and $r_{\text{Spread}}$. To obtain the $r_{\text{Spread}}$ you simply take

$$r_{\text{Spread}}^{(t)} = r_{I}^{(t)} / c^{(t)}$$

This is the "adjusting it against spread dv01" that you have observed.