Ability Scores
Your ability scores can't be right - without a racial bonus (or a feat as a Variant Human), there is no way for an ability score to be 19 as a starting character.
Saving Throws
It appears you've conflated saving throw proficiency with stat bonuses. Being proficient in a type of save does not increase your stat at all times, it only adds your proficiency bonus when an effect forces you to make a save of the type matching the stat.
Backgrounds
Backgrounds are important mechanically, at least for determining starting proficiencies - each background provide two skill proficiencies, plus a combination of any two language proficiencies and tool proficiencies.
They also provide a background ability (Acolytes can get free spellcasting, Soldiers have some reputation from their battle prowess, etc). Some of them are more potent than others, but all have their uses.
As for your desired background, there are quite a few appropriate choices in the PHB. Acolyte or Hermit would work if the monk part is more important, while Guild Artisan would work for the weaponsmith part. Note that it is also permissible to adjust the backgrounds, as long as you take two skill proficiencies and two tool or language proficiencies.
Skill Proficiencies
At the very minimum, a character should have four - two from class and two from background. Some races gives extra, and some classes let you pick more than two.
As a Tabaxi, you should have six - Stealth and Perception from the race, two from your background, and two from Fighter.
Armor
Heavy armor gives you a static AC; your ability scores do not affect it. Chainmail, for instance, is always an AC of 16. A shield would bump it to 18, and better armor could be even higher.
That said, your proficiency bonus is not added to AC by default. There may be feats or class abilities that can temporarily apply it, but it's not a full-time thing and certainly not at 1st level.
Commentary
While creating characters is a decent exercise, and gives you cause to write questions like this, it can only be safely undertaken as a thought exercise. Almost all DMs who use rolled stats will want to have them rolled where they can see, while many use point buy instead. There may also be setting-specific cultural elements (or even mechanical world design elements) that influence race and class combinations.
Best Answer
No, it's added to your score first.
Your racial modifiers add (or subtract, in a few weird cases) to the ability score, which you use to find your ability modifier.
So, like, if you have that Con 15 and your race gives you +2 to Con, you have a Con 17, which you use to get the +/- that gets added to every Con roll.