Convert Ecliptic to Galactic Coordinates
Probably it's easiest to use some package for this, for example, if you use Python, you can use astropy:
http://docs.astropy.org/en/stable/generated/examples/coordinates/plot_galactocentric-frame.html.
Or if you are interested, here is a description on how to do it by hand, including the propagation the uncertainties:
http://adsabs.harvard.edu/abs/1987AJ.....93..864J

Of course, you also need radial (line-of-sight) velocities, but I don't know if that is in the ATNF catalogue...

Edit:
About the radial velocities: we only see the pulsars move across the sky and know the distance, so we get two velocity components perpendicular to our line of sight, a $v_\alpha$ and a $v_\delta$, from the proper motion in Right Ascension ($v_\alpha=d\cdot \mu_\alpha$) and the propermotion in Declination ($v_\delta=d\cdot\mu_\delta$).

Probably, it is useful for you to convert these proper motions from $\mu_\alpha, \mu_\delta$ to galactic longitude ($l$) and latitude ($b$) $\mu_l,\mu_b$, so you can get the velocities in cartesian coordinates: $v_y=d\cdot\mu_l$ and $v_z=d\cdot\mu_b$. This way, you have at least the two velocity components perpendicular to our LOS. Oh, and you maybe also want to subtract the rotation of the sun around the MW center.

There are ways to estimate the radial velocity sometimes: if the pulsar lives in a binary system, a spectrum of the companion maybe. Or a very crude way would be maybe something like http://adsabs.harvard.edu/full/1977ApJ...216..842H... But I don't know much about it.