I was going to set up a Dark Elf character. He was going to be a non-aggressive religious type. He was born in Skyrim, but had always meant to get around to seeing his ancestreal(bad spelling) homeland. He was brought up in the religion of the Nine Divines, but has always been curious about other religions. He will fight when he has to fight, but would prefer to read a good book. He found himself in jail because he happened to fall in lust with the wrong woman and the woman's husband did not approve of his actions. My question is which class would best fit this character description?

How about a Bard?

I can't see a bard being all that religious, but it might be worth trying.

I'd build a custom character and call his class whatever you want to name it.

For example, I played one primarily non-violent Dunmer who was raised in Cyrodiil in a mostly Dunmer ghetto, and learned to despise Imperials. Eventually, he was arrested for making radical Dunmer separatist speeches to a small but growing group of supporters. Much to his surprise, he was taken to Vvardenfell and released. With the implied threat of reprisal for not following orders, he dropped off the package, but refused to follow the Spymaster's orders. That night, he was attacked by an assassin, whom he obviously assumed was hired by the Imperials because of his refusal, and fled by night to Telvanni territory. He spent the rest of the game running from the Imperials, never staying more than a day in the same place, without realizing that the attack was not sanctioned by the Empire. Eventually, he headed to the Mainland (TR project mod), and took up residence in the Port Telvannis area where he felt safe from Imperial pursuit.

He fought only when necessary, and was primarily based around Speechcraft, a charismatic personality, and a bit of magic, including Mysticism, Enchanting, and Healing. Class was custom, called "Prophet", but several other class names would probably fit.

I've played several characters who were "non-aggressive", but were quite willing and capable of defending themselves, including an avid Imperial supporter who did most of his actions "for the Emperor", but was an Archaeologist by profession (another custom class).

Just in terms of gameplay, a pilgrim could be more difficult for avoiding killing because he'd start with 20 hand to rather than 30 (whilst his other melee skill i.e. short blade would be 30. It would rise more quickly than it would for a healer, but you get started, hand to hand rises faster than any skill in the game, possibly besides alchemy if you bulk buy some marshmerrow and saltrice. That said, this won't be an issue at all if you make a custom pilgrim.

Do you have any ideas about how your character became a healer or pilgrim. What did he do when he grew up? For a pilgrim, he may have simply started travelling, perhaps to meet followers other religions, after learning a few essential skill and gathering some essential equipment. A healer from Skyrim (within the Nordic culture) has probably had formal training at the Mages Guild, College of Winterhold or somewhere like the Temple of Kynareth.