I'm going to start a new character soon, but want to try something really different and have settled upon the idea of a character who focuses on restoration (not just healing spells, but also the offensive spells against the undead). From a role-play perspective, I thought of playing as a Priest of Arkay. How could I obtain the robes though? Is there a quest which allows me to become an actual priest? How could I role-play as a priest, whilst still having a logistical reason to quest and whatnot? What other skills besides restoration could I use?

The standard 'priest' robes are Hooded Monk Robes. You can get a free set off a dead priest at a Talos shrine a littke bit to the west of Riverwood where you start the game. Alternatively you can buy Monk Robes from Radiant Raiment in Solitude. If you want an amulet of Arkay, you will occasionally find them at Radiant Raiment, or on dead Imperial Soldiers. I'm sure there are other ways of getting them too.

There is no quest allowing you to become a priest per se, but you can try the 'Book of Love' quests for the Temple of Mara in Riften. Also there are numerous other quests and objectives you can do for the Nine Divines, just talk to priests at the Temples and Halls of the Dead.

As far as roleplaying is concerned, you could view the Draugr as lost souls in need of eternal peace, and fight them in the numerous dungeons in the game. I'd imagine a priest would be fairly pacific towards the living though. So main skills might be:

One-Handed (Maces) or Two-Handed (warhammers)

Block (blocking, duh)

Restoration (Healing, Turn Undead, etc.)

Illusion (Calm)

Speech (Persuasion/Intimidation/resolving conflicts)

Alteration (defensive spells, shields)

Any other skills your character might have from life experience prior to becoming a priest (maybe he was a lockpicking thief before 'reforming'?!)

I was thinking of restoration and archery, using crossbows after having sided with the Dawnguard.

Good idea. If you do do archery, stick to crossbows though. I don't think bows really suit a priest as such. Crossbows are more crusader-ish.

I've wanted to try a crossbow only character, so I felt it fit here.