Sometimes you won't know until you try it out yourself. I made quite a few characters for experimenting when I started out - happens in any RPG games I play, so I don't really count those, they're all level 10 or so. My first serious character was a sneak archer, level 48 but it got a little... too easy. My savegames became corrupted then so I was forced to start over.
My second character was the canonical 2H Nord warrior, I intended to make him the 'hero' of Skyrim, so I did good things all around - joined the Companions, become Thane of three holds, helped people. Also sided with the Empire just because I liked their costumes better - you could see faces!
And then I went ahead and finished the main quest at level 40. After I finished the main quest, I continued doing side quests and exploring for a day or two. Suddenly I completely lost interest in him. I had a story for him too. He started out good, and then once he learnt that he could defeat the evil no one else could, lust for power started to consume him. But I never got that far. I just lost interest, which is werid because I barely explored the world at all. I've never been to other cities aside from Solitude, Whiterun and Riften.
Now I started another character, a complete opposite. She's a Breton (they're small), have no morals and in it only for the money. She's avoiding the main quest completely - no dragons, no shouts. ICly, she escaped from Helgen, and she went to warn the Jarl. Once that was done, however, her job is done - as far as she is concerned, anything else after is the city's problems.
I'm having a blast with this evil character who's an aspiring assassin, and she has no qualms about killing anyone who says the wrong thing. (Ideally everyone in Skyrim's an ass, but as much as I like to wipe out an entire city household by household, ghost cities are not very fun). My point to you is, even if you're not an alt person, try to set aside room for 2-3 characters to see the different parts of the game. Unless you:
1) are an OCD completionist;
2) really dislike the game in general;
3) are the kind as Kasperio mentioned - beat the game and you're done;
then there can always be fun in alts. A few tips for you. The main quest is no different from any of the 'story' quests you come across (quests that don't appear under Misc section). It's only the main quest if you let it be the main quest. Once you treat it as just a 'quest', suddenly there are so many 'main quests' in Skyrim.
Set some small goals for your next character. A little RP is always a great idea, though not necessary, but I find that having a simple backstory for my character to justify his or her actions make it more fun. Try out different perk trees - a few are really meh, but instead of being a DPS powerhouse (the usual 2H, 1H, etc), try quietly poisoning people to death when they aren't looking. Or be a master of punching!
I can't really tell you what to do, it depends on what you like. But you can always find something new and completely different to do from your previous character.