I was thinking of making a nobleman or chef or some kind of non-combat focused character. Like, he'd let his companions do most of the damage and just use illusion for crowd control. How do these type of characters work?

Not well enough, I know people do it but for me this series has a long way to go before non-combat characters appeal to me. I think they potentially could, but at the moment combat is where it's at.

My idea was to create a crafting focused character, who makes strong gear for his companions, and maybe uses illusion magic for crowd control or conjuration for summoning familiars. Could you play the game solely using conjured familiars?

I have played a few characters who did not specialize in combat skills: a miner, a tailor, an entrepreneur. These characters mainly leveled up crafting (in the case of the miner and tailor) and Speech (in the case of the entrepreneur). Their combat skills tended to level very slowly, as these characters usually ran away from enemies. I thought all three characters were very fun characters to play.

Non-combat builds were a possibility, even before patch 1.90. Now, with the legendary skills system, you can have such a character reach a fairly high level, without deviating from your original plan.

At lower levels with weaker summoned creatures it's more difficult for them to stay alive, but can serve a purpose as distraction while you get away. At higher levels with atronach's they give a good account of them selves & then the Daedra Lords (if i recall the name correctly) are badass & when you can summon 2 creatures, it's amazingly good.

Illusion is one of the most powerful schools of magic, maxed out you can control most fights.

I'm playing as a Khajiit merchant who was separated from his caravan and wrongly caught up in that Imperial/Stormcloak thing. He's just going around, gathering trinkets, and selling stuff mainly. Unarmed combat mixed with maybe some illusion .

I made a Breton healer type character who carried a shield and just healed her companions. She only took the first block perk and the rest in enchantment, restoration, alteration, and some illusion. Her role-play was that she was just to physically weak and cowardly to engage in combat. I set her limit on smithing at 60 because of low strength.

I only did the main quest and few side quests with her. I didn't really need Illusion a whole lot once you get your companions well equipped it isn't necessary, nor is Conjuration for that matter. There are staves you can get to conjure, like the Sanguine Rose, even early. It got to the point once I had Jordis equipped with fortify health and stamina gear well smithed that she just killed everything and I barely had to even heal her or do much at all.

Yours sounds like it isn't into the MQ, so I suggest getting Janessa if you are on pc and can remove the auto equipped hunting bow thus making her dual wield.

I think it can be done , but you could only get so high in level. As previously said, focus on all crafting for your companions and magic except destruction. Perhaps take the ritual stone as a just in case.

My Bard character was mostly non-combative. She traveled with her troupe (Lydia, Vilja, and Meeko), and specialized in Illusion, Restoration, and shouts. The troupe would rush into battle while the bard stood back and supported them with healing and crowd control. She did keep a dagger at hand in case she got cornered but for the most part avoided combat when possible (her melee skill was pathetic really). She also made use of magic staves in a pinch (for bringing down dragons, etc). If you have a tank to fight for you, its totally doable and was an interesting diversion from a typical playstyle.

She got to level 40 before finding an interest in Destruction magic and was beginning to go offensive more often. Shortly after that I had to reinstall Windows and I ended up ditching the save. She had a really great run though.