Where do I find Arvak's skull in the Soul Cairn?

After you pass through the "gate" where you met the questgiver, go EEEN (about 70 degrees right from N), it's a long walk but eventually you will find a shrine-like structure which is basically a roof supported by 4 columns, it will be illuminated from within in purple, the skull will be at the foot of a pedestal at the shrine's center. The shrine is due east of the entrance to Valerica's prison.

Here's a video guide.