Here is the traditional way. I think this is the kinematics of your arm, but am not 100% sure.
Here are the DH parameters and transformation matrix:
DH Parameters for the anthropomorphic arm with spherical wrist $$ \begin{array}{c c c c c} \\\hline \text{Link} & a_i & \alpha_i & d_i & \theta_i \\\hline \\1 & 0 & \pi/2 & 0 & \theta_1 \\2 & a_2 & 0 & 0 & \theta_2 \\3 & 0 & \pi/2 & 0 & \theta_3 \\4 & 0 & -\pi/2 & d_4 & \theta_4 \\5 & 0 & \pi/2 & 0 & \theta_5 \\6 & 0 & 0 & d_6 & \theta_6 \\\hline \end{array} $$ $$ {\Large A}_i^{i-1}(q_i) = {\Large A}_{i^\prime}^{i-1}{\Large A}_i^{i^\prime} = \begin{bmatrix} \\c_{\theta_i} & -s_{\theta_i} c_{\alpha_i} & s_{\theta_i} s_{\alpha_i} & a_i c_{\theta_i} \\s_{\theta_i} & c_{\theta_i} c_{\alpha_i} & -c_{\theta_i} s_{\alpha_i} & a_i s_{\theta_i} \\0 & s_{\alpha_i} & c_{\alpha_i} & d_i \\0 & 0 & 0 & 1 \end{bmatrix} $$
Here are general instructions how to compute Jacobian:
And this should be the Jacobian for your specific arm:
Curtsey of Siciliano, Sciavicco, Villani, and Oriolo.
Hope that helps.