Play a ... COMMONER. An average Joe. No fancy spells and groundbreaking swings, just trying to get by.
A few details / tips, you might find useful these:
- RACE: not important, for obvious reasons Nords are favourable. But if you haven't played Khajiit, they can be add an excellent RP perspective to that: they're not allowed behind city wall's, so you have to avoid the 5 biggest settlement. This brings a ton interesting aspects to the game - I strongly recommend that option.
- DAILY ROUTINE: basically earning money in two ways: [a] doing pity quests & [b] crafting / collecting things that don't increse any skill.
[a] typical "take this bottle of wine to the Jarl" / "deliver a letter to my son" / "persuade X to do Y" quest.
[b] these are: Cooking, Mining / Smelting, Tanning, Woodcutting, and gathering Vegetables / Ingridients /Wheat.
At the end of the day, you get septims for these. Of course, you can have additional sources, like selling stuff you loot / find.
- OUTFIT: normal, everyday boots / clothes / hats that almost every non-combat NPC wears. As for weapons, well, they're more like tools: Pickaxe and Woodcutters Axe. You need them for work anyway, maybe a little dagger could be reasonable, but armor / shield / helmet / etc. stricly not allowed. You can have different sets for different workes - if you're working in the mine, pick the Miner's clothes for the perfect experiance.
- CHARACTER DEVELOPMENT: [a] stay at level 1 (optional) & [b] don't do MQ & [c] appoint additional challenges [d] roleplay
[a] its an interesting way of playing (a guy in the forum doing this for months): no perks, no health / stamina / magicka gain. But you're just a commoner, why you need those? Perks can make a big difference, but your skills are developing anyway, so don't worry. Oh, one sidenote: Death is Death could be a great other self-made challenge besides that. If your character dies, delete your save game.
[b] its a must, after Helgen just stop doing that, don't even report to Balgruuf! Which means you're not a Dragonborn (technically yes, but I hope you see my point), and you literally don't have to fight with dragons. (And that's sure from technical staindpoint too).
[c] doing aforementioned things will mainly increase two things: the Speech skill and the amount of money you have. So the next big challenge: be a merchant. As a merchant, you need a bodyguard. Hire one for 500 septims. Buy a horse, if you want. Looks familiar? (There is a random encounter just like this).
After that the next challenge: be a Thane in every region you can reach! Because doing a lot of favours for the people, you'll very close for that. But to achieve it, killing is inevitable - but you can let your follower do it! Unfortunetly, you have to go with him / her to dangerous places, just be creative: buy invisible potions / muffled boots / enchanted jewelry just in case of trouble.
[d] you have basic needs: eating / drinking / sleeping etc. Roleplay that. Set a carry weight limit, the default is too much. If you need something (cloth / ingridient / etc.) collect the money for it! There are cold out there, don't go out without warn clothing! Or did you drink too much the tavern? Punch that unsympathic, annoying dude in the corner - and do your jail time for that. Everybody can make mistakes.

- PURPOSE: with that build, you can beat "filler" quests which ones are otherwise don't make sense. In other playthroughs you're a Dragonborn, a life-saving hero, not a damn courier! In here, you just an everyday guy looking for honest ways to make his living.
Plus you'll have a lot of little things you have to deal with. One example: if you choose to roleplay Khajiit, you won't have a house to live with (since every purchasable house are located in the 5 biggest city). So you have to rent a room every night, everywhere you go. But what if you help to Faendal in Riverhood (his quest is typical filler one)? He'll be your follower - so you can live in his house.
Or as a Khajiit, you should trade a lot with their caravans. If you have to go somewhere, and you don't want to go alone, yet you can't afford a follower - why don't you wait for them, and go with them? They'll protect themselves (and actually, you) from every danger while you can reach your destination safely. All you have to know is their route and wait for them.
- CONCLUSION: I know this doesn't sound interesting first, that's why I went to the details that much. If you really are for roleplaying and trying every possible aspect of the game, I think its THE way to see those sides that you're normally never. Basically through your gameplay you will be wandering from area to area, harvesting / mining / etc. and doing local sidequests. If you see the big picture, a lot of micromanagement is sufficient to move towards your next goal. This playing method is parallel with Skyrim's respawn system (time), so you can farm every area again and again.
Damn, sorry for the long post, but anyway, I hope that helps!