First, let me tell you about the pitfalls of Lineweaver-Burke plots. The tranformations of your data (i.e. taking the reciprocal) distort the error structure of your data and can lead to erroneous fits. It is better to directly use nonlinear fitting techniques to find kinetic parameters. These techniques do not require transformation of the data and thus do not distort error structure.
Second, let me answer your question. The classical Lineweaver-Burke plot puts inverse reaction rate on the $y$ axis, i.e. $1/V$. Your question is how to find $V$ from absorbance data. The answer is, you don't need to. Let $A_i(t)$ be the absorbance data for tube $i$ as a function of time. What is $V_i(t)$ for that tube? We won't know unless you know the molar absorption coefficient of the substrate (or product -- I'm a little unclear from your question what you are hoping to detect by measuring $A_{\textrm{530 nm}}$). Assume that it is the product that is absorbing, and that it has a molar absorption coefficient of $\epsilon$ and that you are using a photometer with path length $L$. Then the reaction rate for the $i$th tube will be
$V_i(t)= \frac{1}{\epsilon L}\frac{dA_i(t)}{dt}$
This means take the slope of the absorbance measurement and divide it by $\epsilon$ and $L$ -- numbers you don't know -- to get $V$. But the key assumption is simply that $V \propto \frac{dA}{dt}$. Try making up any numbers you like for $\epsilon$ and $L$. Find $K_m$. Then make up new numbers and find $K_m$ again. Did the result change?
