While it's POSSIBLE to get everything you want into a build, it'll be spreading your stats very thin and require a big pile of raising stats with perk points. Thinking about what you want most at the end, which is a heavy weapon user with high carrying capacity and leadership perks, I would honestly just drop the point into Agility it would require for the more ammo-friendly Rifleman perk, which you'll be able to unlock on the way to getting demolitions expert anyway. With that as a premise, I would personally go for: -Strength 5 (bumped to 6 for Strong Back) -Perception 5 (For Demolitions Expert) -Endurance 4 (gotta have health, especially without Power Armor) -Charisma 6 (for Local Leader) -Intelligence 5 (for moderate experience gain and the Scrapper perk) -Agility 1 -Luck 2 (for Scrounger) Perks: Strong Back, Armorer, Gun Nut, Rifleman, Demolitions, Local Leader, Scrounger, Heavy Weapons, Lone Wanderer, Caps Collector, Lifegiver. Playstyle: Even with four points in Endurance, you're still going to be a little bit squishy. Emphasis on early game should be getting Armorer and Gun Nut to improve your gear, Strong Back for carrying capacity, and Lone Wanderer for further raised carrying capacity and damage resistance. Your Charisma boosts should allow you to buy, sell, gear, gear up, and win persuasion checks. Late game, you'll have extremely high carrying capacity, sturdy armor, grenades to lob over fortifications, and Heavy Weapons.

I am currently stuck in my RP as what I call a Tiamat:

She is a human version of the legendary dragon (I researched her and found out she is also capable of appearing as a very attractive woman), capable of all sorts of breath attacks.

Her primary weapons are mid-level Destruction spells (Firebolt, Iceball, Lightning Bolt) and shouts.

Her main shouts are supposed to be: Fire Breath, Frost Breath, Storm Call, and Elemental Fury.

However, she cannot get the first word of Fire Breath without joining the Companions, and I don't want her involved in that path (she is supposed to be College of Winterhold, Thieves Guild, Dark Brotherhood.)

Is there some way she can get into Dustman's Cairn to learn Fire Breath without becoming a Companion?

I dont think so, but note that the dps/usefulness of most shouts is related to their cooldown. So Frost and Fire Breath booth have less dps (damage per second) when used with more than one word because the cooldown increase makes the dmg actually less over time then when used with 1 word only.
Even if you have 80% shout cooldown reduction, it's still the same.

Thanks - I now know that one word of the shouts will serve her well.

And I know where to get "Inferno" from Fire Breath, and can get it.

I guess you're going for shout cooldown reduction? Know that there is an easy way around Alchemie and stacking:
(Main Quest Spoiler) During the quest "The Throat of the World", you will face Paarthurnax and he will ask you to breath fire at him. If you take the word from the wall and fast travel away without doing so, you will receive an invisible "buff" that resets your shout cooldown timer every 3-5 seconds. It's a bit inconsistent (actually from 1-8 seconds), so I wouldn't call it overpowered, but it's crazy effective and easy to get.

Just in case you still want to get all three words of Fire Breath, there's this little tidbit from UESP's page on the shout:

If none or only one of the words of the shout Fire Breath is known when you first approach Paarthurnax at the Throat of the World, and you have a dragon soul in stock, you can use the soul to learn the newly found word on the wall before Paarthurnax unlocks it for you. This results in the "soul" from Paarthurnax teaching you the next word of the shout. This way you can learn two words of the shout at the same time.

If you do this when you have no words for Fire Breath, the next word wall for Fire Breath encountered will not unlock the final word. All three locations will still need to be visited to unlock the final word. If you have unlocked one word, you do not have to visit the final location; however, it will still run the animation as if you were learning the word.

So, if you pick up the word for Fire Breath in Sunderstone Gorge and then use this trick with Paarthurnax to pick up the next two words at the same time, then you should be able to pick up all three words this way without having to join the Companions.

How do i post a Build Here? I had trouble posting my last build the End Maker on the  Character Build Section too.

First of all you have to read and understand the Character Building Group Rules. There is everything listed that you need in your build to post it (successfuly) into the building group.
You should also read the Guide to Characterbuilding. Additionaly, you can take inspiration from other builds.
I recommend putting the build into the workshop while you're working on it. To start a new discussion there go to the Workshop and scroll down until you see a button called "Add discussion".
It is important that you title your work properly, like this:
WiP Characterbuild: ???
Your build must have a gameplay section, so people know how to play it, and a perkspread. Read through some of the builds, if you're having trouble.

You can create your perkspread here and put the link into your build (you can also ask in the Art Goup Services Request Tread for a perkspread with picture). If you need any help with gameplay, -mechanics or something else, you can ask in the Character Building Help Desk or in the Tipps & Tricks Group , respectively.

When builds have an abilities section, where do people usually get the little icon-like pictures they have accompanying the ability? I've been scouring Google images and can't find anything like them to use for my build. Maybe I'm just not searching the right things?

Ability icons from games like WoW or Guild Wars 2 work very well due to their variety. Here is a link to the GW2 icons, just click through the sections :).
https://wiki.guildwars2.com/wiki/Category:Skill_icons