Step 0: Keep early bow weapon, we will be using this until we can cast Bound Bow
Step 0: Buy Bound Sword from Farengar when you start the early main quest to retrieve the Dragonstone at Bleak Falls Barrow
Step 1: Buy Eidar Cheese Wheel and Ale from the Bannered Mare
Step 2: Get moon sugar from the Khajit caravan that appears outside of whiterun every few days
Step 3: Make Elsweyr Fondue with those Ingredients and hold onto it (bannered mare has a cooking station off to the left side of the main hall)
Step 4: Go to bleak falls barrow and clear the place out up to the Wounded giant frost spider, do not kill him. He can't reach you, but will allow you to level non-damaging combat magic.
Step 5: Eat Eslweyr Fondue, and recast bound sword over and over again until you have Conjuration 50 (takes maybe 5 minutes), make sure to perk Adept Conjuration, as this will cut the bound bow cost in 1/2
Step 6: Leave the Barrow from the way you came in, we will be back after we have our prize
Step 7: Head back to White Run stables and take the carriage to Winterhold
Step 8: Become a member of the college and hunt down Phinis Gestor, buy Bound bow from him. (or you can lookup where to find the tomes, but they are not exactly easy to retrieve at this stage)
Step 8b: If you have a problem casting the spell to join the college (eg: You get stuck with Fear as your entrance exam) go to Riverwoodd Trader in Riverwood, he will have 2-3 Eidar Cheese wheels... and again, make elsweyr fondue to get your mana to 200 to cast that pesky spell to get entrance to the college.
You are now ready to mow down anything that comes your way for the rest of the game, return to Bleak Falls Barrow
Notes:
-In addition to perking the Bow line, enchanting to get +bow damage items is also real nice. (especially at 100 enchanting, when you can put a +bow enchant, and then add some other enchant to be useful)
- As an assassin, at some point soon you are going to want to get "Quiet casting" in the illusion line. You need Illusion 50 and 4 perk points. You can power level through to this too. Find Drevis Neloren at the college, buy muffle from him, AND complete his college quest to cleanse the mana wells. This will give a temporary +2500% mana regen (heh, yeah clearly a bug). Eat elsweyr fondue again. Cast Muffle repeatedly until your illusion is at level 50 (again, this is only a couple minutes of work).
-One final word, we go to the college because Farengar will not sell bound bow until after Dragon Rising, even if he technically should sell it to you because your conjuration is high enough)
Personal Note: Quite frankly arrow management is beyond annoying, this is why I love bound bow.