It's totally doable on default difficulty. I had one character who was a no-sneak archer, who got by fine with just kiting. Once you get the stagger perk, archery can be quite powerful and staying out of reach is not that hard. Granted, most of my archers are Khajiit, so they do have their claws as a back-up and use them sometimes in the beginning while they are working their archery skill up to get the stagger perk.
But once you have that perk, there should never be a need for a melee weapon, if you are careful not to get cornered and always have a place to run to. I don't use the slower bows that do more damage, but stick with faster bows like hunting and foresworn. With the stagger perk, it is more important to hit frequently than to do a lot of damage per hit. That, plus a little sprinting will let you fight a lot of enemies without ever having to switch to melee.
Stamina is an archers best friend, both for the bow zoom/slow time perks and also for the sprinting that you need to do on occassion to put some distance between you and your enemies. I am not a big Skyrim fan overall but I do like the way they handled archery. For added fun, turn off the crosshair and use the instinctive method of aiming your bow.