How should I build a character that exclusively uses bound weapons?

Generally speaking, as a weapon-using character, there's not going to be much difference between a bound weapon and a physical one. So, to answer your first question, the other trees useful to your path will be the same ones that are useful to any bow/battleaxe-wielding character.

As for armor choices, you might want to take a look at this question about warrior armor types, or this one about two-handed warrior mages.

Finally, since bound weapons are roughly equivalent in strength to Daedric weapons, I see no reason why using them on Master difficulty wouldn't be viable.

You should fully upgrade the bound weapons part of the conjuration tree, and also upgrade 1-handed, 2-handed, and archery as these all contribute to your bound weapons. In my opinion, bound weapons are the best weapons in the game as they have the equivalent damage to daedric weapons and are weightless so it uses less stamina to power attack.

As far as the armor goes, if you want to keep with a magey type feel I would suggest Oakflesh, Stoneflesh, Ironflesh, or Eebonyflesh, and get the Mage Armor perks and use robes for good magicka regeneration and the decrease in cost. If you want the best armor rating possible, use a set of legendary daedric and enchant magicka regen on it .

Bound weapon logic: Bound battleaxe: 25 damage, weight 0. Tome cost 356 from Phinis Gestor
Ebony warhammer: 24 damage, weight 25. Need a high level to find, buy, smith.
Daedric battleaxe: 25 damage, weight 25. Cannot find, need REALLY high smith skill.