Oliver Queen was a spoiled rich kid, living in the Imperial City. His father, Robert Queen, owned Queen Consolidated, one of the most powerful companies of Tamriel. Oliver's mom recently was killed by the Dark Brotherhood, and his father thought Oliver could use a business trip to Skyrim. But on the road back to Cyrodiil, the group was ambushed by the Imperial Legion. Mercer Frey feared Queen Consolidated would overshadow the Black-Briar family, and hinted general Tullius the group were Stormcloaks in disguise. The Legion killed everyone that resisted. Oliver was the only survivor, and was brought to Helgen.
After escaping Helgen, Oliver travels to Solstheim to deal with the loss of his father. He trains there to become the best archer, and save Skyrim by destroying the Dark Brotherhood and the Thieves Guild as Green Arrow
The Green Arrow deals his damage as an archer, using alchemy to create poisons for specialized arrows. When enemies come too close, he bashes them with his bow, and take more distance. With light armor, he will be quick, but also somewhat protected. Smithing will let him make glass arrows, to resemble his name of Green
Arrow. He won't be enchanting neither his armor nor his bow. He also frowns upon using any kind of magic, so no spells or shouts.
He will wear a green hat and glass armor. Jewelry is optional, but they won't be enchanted! Once he leaves Solstheim he won't use any other bow, just the one he has used on Solstheim. All arrows can be used, but glass arrows are preferred.
As soon as you arrive in Skyrim again, head to the face sculptor to get a haircut and beard and to give yourself a 'mask' of green warpaint. Then, destroy the Thieves Guild and the Dark Brotherhood. Dawnguard and Companions questlines can also be followed. The main questline and Dragonborn DLC require the use of shouts, so they can be followed, but use shouts only when there is no option.
For roleplaying purpose, play this build as an Imperial. This will give you a +5 boost for block, so your bashes are stronger at the beginning. Take the Serpent Stone for one very strong poison on your arrow once a day. Spend no points in Magicka, 40% in Health and 60% in Stamina. You can have followers, but only followers that don't use Magicka.
EDIT: Since key members of the thieves guild are essential, install a mod to make essential NPC's non-essential.