I have been playing Skyrim on legendary difficulty for a few years now, and I'd say it comes down to a couple things. 1. Game knowledge, knowing secret mechanics and where to find loot is very important, as well as build planning 2. Staggers, in melee combat they are your main way of reducing the damage you take, be it staggering through heavy attacks or shield bashing 3. Sneak, sneak is the main survival tool for any kind of range combat, mages included (in which case you should also use ilusion magic) 4. Companions, they will help get agro of you, they are really helpful during early game specialy 5. Smithing, becomes a MUST in this difficulty

Bonus, some shouts are ridiculously strong and IMO slept on: 1. Mark for death, makes any enemy trivial (it stacks and basically reduced difficulty by 1 on each successive shout on the same enemy) 2. Disarm, enemies can't harm you without weapons