Yknow, I will actually give Skyrim credit and say magic builds are the most fun. Dunno why, but there's a sense of progression there you don't really get from the other two.
Having said that though....are you asking if it's doable without dying or just doable in general? And on what difficulty?
Issue with a mage build is you'll quickly find yourself with 0 magicka and you're dealing abysmal damage compared to the other character types. You can NEVER have enough magicka, but the moment you focus solely on magicka, obviously you're a oneshot.
And on top of that, you can't really level your skills as fast as you'd like. Alteration, conjuration and destruction can typically only be leveled during combat. For example if you're sneaking up to a group of enemies and you put on your magic armor with alteration, that doesn't actually count towards leveling your alteration because you're not in combat yet. The result is you'll find yourself spamming spells as you travel to level their skills. Destruction also levels DAMNED slow and it's DAMNED weak; the only thing that makes destruction at all viable is Impact, which lets you spam stagger on enemies. (but they still take ages to kill)
So tbh, I can't say if mage was actually fun to PLAY or rather it was fun having all the different toys to tool around with and fun from a perspective of character progression, since the character was definitely weak so every single perk was a welcomed improvement I desperately needed, with the character slowly going from "pathetic" to "pretty damn good." Been a while since I played one. With mage, you will be spending some time running from enemies and waiting for your magicka to regen, and assuming you keep putting points in magicka because holy crap do you need it, you'll die quickly so you're constantly on your feet. But it's sort of an opposite problem in that now the game can sometimes be too hard. I actually distinctly remember a fight vs a Volkihar vampire and it's thrall that was the most frustrating **** ever. Why? The thrall used a bow, the vampire spammed magic attacks. A SINGLE hit from either was enough to kill me. This forced me to basically summon something, HOPE it drew fire, then absolutely spam Impact on one or both of the enemies to keep them stagger locked.