Has anyone specialized in Conjuration and Illusions, with melee only as a backup? (and no bows or destruction)

If so, how well did it work? What spells and perks are needed, and which ones can be skipped?

This is one of the most popular mage builds you can find. There are a ton of threads devoted to this very thing. This is very popular among Vampires as well. They are an extremely powerful combination. Some may even consider them to be gamebreaking. I consider them to be quite fun.

Conjuration is just what ever you want. Of course the higher level the spell, the more powerful the summons. casting two Dremora Lords are absolute powerhouses. I usually forego either the necro or the atro side of the Conj tree, pending the character I am playing at the time. But one can easily do both.

The Dead Thrall spell of the Necro side is quite powerful. You can upgrade their default armor and they will use any weapon you put in their inventory.

Illusion is an all or nothing type of skill to be effective the entire game. If one is truly bent on utilizing Illusion, they need to plan on investing in every perk.

I must read up more on this, because the second character I started I decided would be a Conjurer Illusionist. So hard not to pick up swords and bows and use them loads at the start though...

The Breton is attractive race for this, as they start with a Conjuration spell. I think the the thing that makes this type of character the most fun, is avoiding combat all together. Some say...

"That is no fun. Just get someone else to kill for you? Pft...whatever. That just sounds dumb."

Well it's not as easy as you think. Especially if you don't put anything in Health and play on master. Trying to play the game and not ever get hit is tricky.

I made one with one handed as my primary combat skill but was using the bound sword. It was pretty fun. I use other magic skills to back me up but thats my go to weapon.

I've a destruction / illusion Dunmer mage-assassin. That works well.
He is in light armor and uses illusion magic to turn invisible and remain undetected (muffle is enchanted into armor). He places or throws runes a lot. In combat he uses ice spikes or flames. I could be imagining but I'm sure ice spikes seem to be more stealthy than fire. I find I can stand a far whilst dual shooting them with ice spikes and stay undetected.

I've also a conjuration / alteration Breton witch. That also works well.
She uses alteration magic for armor and then conjures a astronaut (or similar), then conjures a battle axe or bow to fight melee style. She also uses potions to support.