Hi, welcome to the guide. I have spent many a long hour playing Ori bot and would like to share my wisdom with the world! As a preface I would like to remind you that I have very little competitive experience, so if you came here looking to make Ori APC comp viable I have two things to say:
We all do.
Take this advice with a grain of salt.
With that out of the way I want to point out a few things going into this build. One, it (pardon my french) does a heck ton of damage. This build gives you the ability to turn AP (which items give at a higher rate) into bona-fide AD. All will be explained as we dive into the details.

Orianna APC is a strange concoction and may not be for everyone, but if you pull it off you will never have to watch Rick n Morty ever again!

Pros

High Auto-Attack Damage
High Ability Damage
Great for Objective Securing
Can Avoid Armor
Solid Early Game Poke
Solid Late Game Overall
Acceptable Survivability

Cons

Difficult Farm
Low Mid Game Auto-Attack Damage
Poor Mid Game Overall
Prone to Invisible Assassins
No Mobility
Difficult to Maximize Damage
Early and Mid Game Countered by Tenacity and QSS
Instantly Dies if Hooked
Cannot Take Sustained damage
Mana-Guzzling Early Game

Just like any "APC", Orianna relies heavily on Nashor's tooth but also finds herself struggling to farm in the early game with her autos. This can be offset by using her abilities which drains mana REALLY fast. The solution to this problem is to build TotG before Nashor's. You will eventually finish Archangels but, for the time being TotG Doran's Ring and your Runes will get you by. On a good day you can get Nashor's and Guinsoo's before finishing Archangel's but I wouldn't count on it as you will build up TotG charges very fast.

When all is said and done your core build is definitely Archangel's, Nashor's, and Guinsoo's but my first analyzation of the build will focus on using the 5 items I listed as the core build alongside Swifties:

Archangel's and Rabadon's provide immense power spikes with the theoretical AP total in this build topping of at a massive 618 AP which, believe me, will be a lot in the end of all of this. (711 AP counting Guinsoo's at full stacks)
Orianna's passive, Nashor's Tooth, and Lich Bane all trigger an additional time off of Guinsoo's Rage for some beautiful damage totals: 1,393
This build is mechanically intensive, to reach that 1,393 damage number you need a lot of things... Guinsoos Rage, max Guinsoos stacks, AND Lich bane before you make your attack
Your normal basic attack still does 295 without anything special but If you kick the mechanics in this build to their limit this number increases exponentially
All these numbers are before the 20% bonus on Ori's passive
These high-mechanics situations make her a challenge to farm with

Now there are a few alternate items you can swap in. just know that you may not do as much damage if you switch out portions of the build.

Abyssal Mask and Hextech Gunblade serve the same purpose, generally. They provide a way to gain health back. These can replace Archangel's or even Rabadon's but will lower your overall damage.
Firecannon is an acceptable replacement to Lich bane. It does mean you will not have as high of numbers but you will get some rand off of your energized attack as well as gain a little more consistency.
Morellonomicon is a hard one to switch in. You can replace boots with it but be forewarned: You have no mobility and will frequently be a sitting duck if the fight begins to move towards or away from you!
Spellbinder can replace boots as well, this is for those really late game damage bursts. The extra AP granted by the Item can contribute up to 320 extra damage as well as give you the mobility to use that damage and escape.

Ill review each run now in bulleted list format:

Summon Aery should be a no-brainer, it boosts our numbers on our autos, abilities, and even our shields. This is a far better choice than comet
Manaflow Band is also an auto pick for a number of reasons. 1, Ori is a mana guzzler. 2, it gives us 7.5 bonus AP. Take all the bonuses where we can as that later translates into roughly 12+ damage in our full-burst autos.
Transcendence could theoretically be celerity but in the end the CR bonus at 10 minutes helps us to continue poking and zoning as well as increasing our CR cap.
If you really want scorch go right ahead but Gathering Storm is by far and away our best option. We are already a late game centered build so why not act like it.
Biscuit Delivery helps us by giving more pots and the 160 max mana can translate into 4.8 AP and that into 7.68 damage in our burst autos.
Given Orianna's lack of mobility, we need the bonus movement speed from Magical Footwear and the effectively free 300 gold doesn't hurt either.

This is probably the most challenging aspect at first until you realize that this build include a TON of mana fixing. Between your runes and TotG you should be able to use your Q as both a poke and a farming tool until you can do some damage with your autos. Simply chuck your ball through the minions while their in a line then return it with your E. Do take care when doing this, however, as you can accidentally put minions so close to death that your own minions might kill them before you can secure your gold. Also as an additional tip, the Doran's ring damage will double from Guinsoo's rage so try not to sell it until you absolutely need the space.

I will restate all of the typical Ori strategies in here but one important note here is that you ARE the ADC replacement, so act like one. Make sure you are always putting out your damage but also don't initiate fights with your autos.

Send your ball into the fray at all times possible. If you can land a ult at a good time you can hook the enemy team while they are fleeing or even save an ally from a charging Darius.
Your E is solid but remember, you can cast it on allies
Your W is your best damage output ability so use it wisely
Most importantly, try your hardest not to feed. Your strong poking nature will help to deter the enemy laners but you are no superman and your shield WILL be broken and you WILL die. Learn when to peel and hecking peel.

The hardest part of this build is orchestrating the copious mechanics found in this build. To maximize the damage follow the following steps:

Try to always basic attack minions, the Guinsoo's effects contribute 15% of your damage as well as doubling every other attack's on-hit effect which is 50% OF YOUR DAMAGE. so make sure to do this yeah?
Pay attention to which attacks are triggering Guinsoo's. If you figure this out you are most of the way to reaching that beautiful 1393 damage number.
Once you know when Guinsoo's will trigger you can now begin to integrate Lich Bane. Because Ori's Q is on a 3 second timer before CR and Lich Bane can activate every 1.5 seconds, you can hit half of your on-hit-doubled autos with Lich Bane.

If you can accomplich all of this while also managing Ori's abilities you can deal 1393 damage to a person of your choice every ~2 seconds.

As a final footnote, against tanks you will gain a big bonus to your damage if you continue to attack the same person over and over again. With Guinsoo's this damage can be a whopping 751 additional damage on each doubled hit.

Ori ADC is not for the feint of heart. If you can juggle all of the mechanics you can output some crazy damage, but know that you are still a mage with a decent amount of AP. Everything you do will hit like a truck if you play it right.