Both the forward kinematics and inverse kinematics aren't too difficult, but always a little tricky for parallel manipulators like this one. Consider the configuration in this diagram.
The forward kinematics first involve solving for the position of the joint where you hold the pen from each motor joint separately and then equating the two.
$\begin{bmatrix} a - b \cos \theta_2 - c \cos \beta \\ a \sin \theta_2 + c \sin \beta \end{bmatrix} = \begin{bmatrix} e \cos \theta_1 + d \cos \alpha \\ e \sin \theta_1 + d \sin \alpha \end{bmatrix}$
This gives you two equations in two unknowns, $\alpha$ and $\beta$. After you solve for those angles, simply substitute them back into one of the sides of the above equation and you have the position of the pen given the input motor angles $\theta_1$ and $\theta_2$.
For the inverse kinematics, you start with the known position of your pen (the blue dot in the image), then you compute the lengths of the red lines, $f$ and $g$. Since you know all other lengths you can also solve for the angles in triangles $def$ and $gcb$. From that geometry you can then get the angles $\theta_1$ and $\theta_2$.
So then you can turn each of your desired clock characters into an x-y path for the pen, convert the x-y path into $\theta_1$-$\theta_2$ trajectories, and control your motors accordingly. It looks like the entire linkage is then mounted on another joint that pushes the pen away from the surface between characters, but the logic behind that should be pretty easy to implement.
