Fun things to do in the Elder Scrolls: Skyrim
Become a pacifist
Try to beat the game without killing anyone yourself. Try going the conjuration route and just keep sending waves of atronachs at your enemies. This may not seem like fun on paper but it can be extremely entertaining to take out a room full of enemies and you never have to do a thing. At higher levels this actually makes the game quite easy too. You don't take any damage and you don't have to put yourself in danger so as long as you have magica you don't run in to many problems. I did this in both skyrim and oblivion and even though it makes for a pretty easy run it was still fun from start to finish.
Be an illusionist
The illusion school of magic is extremely entertaining. Its pretty fun to hide out of sight, cast mayhem, and watch a whole city wage war on each other. If you get caught no worries, cast harmony and everything will be fine. And I can't begin to tell you how many times I've cast paralysis on enemies on high cliffs just to watch them plummet to the bottom unfazed. There are tons of things you can do with illusion magic, just be creative and have fun. But remember you can't make it through the game using only illusion magic due to many powerful enemies being immune to it which is a bit of a kill joy
For anyone who may not know what kiting is, it is getting something to attack you then running away but keeping close to it so it follows you to wherever you wish to lead it. I have done this several times in skyrim and the results are usually pretty entertaining. Just the other day I kited two trolls to a farm near Riften and though they got dispatched fairly quickly by the guards, they still managed to kill two civilians who decided to attack them with daggers. This is also fun to do with dragons. Just be aware that anyone can die in these events if they are not essential so be careful where you kite to. I've managed to kite Sven from Riverwood all the way to a giants camp near Whiterun. Oh Sven, we hardly knew ye.
A lot of people steer clear of pickpocketing because they don't feel like getting caught by guards or they think leveling is too difficult. While you do run the risk of getting caught it doesn't actually take too long to level. And once you have your pickpocketing maxed out it can be extremely fun to use. You can poison people and see everyone watch in awww as they just drop and nobody knows what happened. Or you can steal someone's weapons and then fight them and watch them either run or swing away as you easily defeat them. If that isn't enough, you can pickpocket peoples clothes right off their backs!
Create an army
While I'm aware you can only have so many followers, that doesn't mean you can't have an army come time for battle. You may have one human and one animal companion so start there. Then level conjuration to 100 and make sure you get the dead thrall spell and the twin souls perk. You can raise kill 2 enemies of your choosing and then put a summoning staff on there body before you raise them. When you get into a battle your companions will attack and then your thralls will use their summoning staffs and they and the summonings will both attack giving you a six man team which will destroy whatever stands in your way.
Yeah it's fun going through the game being virtuous and good on your first run through. But even then you know you wanted to take out your battle axe when you heard someone make a smugg remark. Well on your next play through make everyone feel your wrath. Your first step into the city just take your sword and charge the first person you see. After that steal everything in sight. Then when you're done move on to the next city and do the same. Just be careful not to do this too early. I did on one run through and ended up getting locked out of Riften. That's not to say that it wasn't still a good time though.
Though it is called a Role playing game, that doesn't always necessarily mean you are role playing. Sometimes its just fun to create a role for your character and just go with it. For example, maybe you want to make a huntsman who is an expert marksman and makes leather armor or a bandit who distances himself from society and whose only companion is his loyal dog friend. A lot of people deem role playing nerdy but it can be a lot of fun if you give it a chance. You can be anything you want, an assassin who strikes from the shadows, a hardened warrior who demands just and see that it is served by any means, or a mage who studies each decision to come to the wisest conclusion.
Sure the Dark Brotherhood has a whole questline dedicated to it but you can still carry out this task on your own. Find a good vantage point and a worthy target, draw your arrow back, and let it fly. Its pretty funny watching the whole town gather around a person you just sniped and if you want to add to the fun you can use a low level bow and arrow with a fear enchantment and just shoot at random targets and watch the whole town run in fear. You can also with the dagger approach. With the 15 times backstab damage bonus in the sneak tree and the 2x multiplier you get from the Dark Brotherhood gloves, you can hit for 30x damage and not many people will resist that. With one hundred sneak skill you can assassinate a target that is right next to someone and they won't even notice.
Dont use weapons
I can't take credit for this idea. I saw a video on it a while back and it was awesome. You pretty much do exactly as the title says, don't use weapons. Running around unarmed can be both challenging and fun once you get the hang of it. There are several things you need to do in order to become proficient at it but it can lead to an interesting play through.
Combine these ideas!
You don't have to stick to just one of these ideas, you can combine any of them together to make a whole new experience. Be an illusionist who doesn't use any weapons, create an evil army, be a pickpocket and have your atronachs attack if anyone catches you. The possibilities are endless in skyrim but these are a few things I do to have some fun