Composition of Lorentz Transformations If a particle is moving in the $x$-direction with velocity $c/2$, then the Lorentz transformation $\Lambda = \begin{pmatrix}\gamma & -\beta \gamma  & 0 & 0 \\ -\beta \gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix}\cosh\ \phi & -\sinh\ \phi  & 0 & 0 \\ -\sinh\ \phi & \cosh\ \phi & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix}\frac{2}{\sqrt{3}} & -\frac{1}{\sqrt{3}}  & 0 & 0 \\ -\frac{1}{\sqrt{3}} & \frac{2}{\sqrt{3}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$, where the rapidity $\phi$ is given by $\tanh\ \phi = \frac{v}{c}$.
Subsequently, if the particle is moving in the $y$-direction with velocity $c/2$, then the Lorentz transformation $\Lambda' = \begin{pmatrix}\gamma & 0 & -\beta \gamma & 0 \\ 0 & 1 & 0 & 0 \\ -\beta \gamma & 0 & \gamma & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} =  \begin{pmatrix}\cosh\ \phi & 0 & -\sinh\ \phi & 0 \\ 0 & 1 & 0 & 0 \\ -\sinh\ \phi & 0 & \cosh\ \phi & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix}\frac{2}{\sqrt{3}} & 0 & -\frac{1}{\sqrt{3}} & 0 \\ 0 & 1 & 0 & 0 \\ -\frac{1}{\sqrt{3}} & 0 & \frac{2}{\sqrt{3}} & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$.
Therefore, the combined transformation $\Lambda''(w) = \Lambda' \Lambda = \begin{pmatrix}\cosh^{2}\ \phi & -\sinh\ \phi\ \cosh\ \phi & -\sinh\ \phi & 0 \\ -\sinh\ \phi & \cosh\ \phi & 0 & 0 \\ -\sinh\ \phi\ \cosh\ \phi & \sinh^{2}\ \phi & \cosh\ \phi & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$.
But now I'm having a bit of a trouble finding the boost velocity $w$ for $\Lambda''(w)$.
Any suggestions?

If you are doing what I think you are doing then, you are trying to get the Boost matrix for an arbitrary direction. Then the way to go about that will be to use the generalized boost matrix (see J D Jackson page547 or http://en.wikipedia.org/wiki/Lorentz_transformation). The the matrix against which you can compare to get the velocity from wikipedia is image. 
In particular, look at the firs row and first column then $\Lambda_{11} = \gamma, \frac{c \Lambda_{12}}{\Lambda_{11}} = v_x, \frac{c \Lambda_{13}}{\Lambda_{11}} = v_y$ and $\frac{c \Lambda_{14}}{\Lambda_{11}} = v_z$
Aside: Are you sure your last matrix is correct? It isn't symmetric.  
