People say you can build out a character however you like irregardless of which class you picked at the start, but that is not entirely accurate. Since you are capped at about a level of 120 if you want to be able to summon help on troublesome bosses, to min/max a character you'd really want to look for the class with the LOWEST stat scores in attributes you do NOT want. Eg. if you do not need Luck, but roll a Sorcerer then you will always be 5 levels behind a class that only had a 7 in Luck. This is one of the reasons the Knight is so good, since he funnels his attributes into ones that benefit the most amount of people.

There is an excellent character calculator available here:
https://sites.google.com/site/darksouls3stats/home
You just enter in your desired endpoint stats and it will show you which class is able to attain those stats with the lowest soul level possible.

Vigor = 17/26/44
This is the most complicated stat; everyone should take at least 17 Vigor, after that the next breaking point is 26, at 27&28 there is a large decrease in value but it flattens out until 44 after which diminishing returns really take over.
Attunement = 35/40
FP runs into a breakpoint brick wall at 35, if you only care about FP stop there, however an extra 5 points will give you an extra slot at 40 so it is your call which you care about; FP or slots
Endurance = 40
Another breakpoint brick wall here, you get good stamina-per-point up to 40 then it hits the brakes hard
Vitality = 40/99
Equip load-per-point is linear all the way to 99, but physical defense gains start hitting a diminishing return after 40
Strength/Dexterity/Intelligence/Faith = 40/59
There is very little reason to push past 40 for any of these unless an item/spell requires it, diminishing returns kick in hard
UPDATE! My methodology for testing INT/FTH may be flawed, I am told they pick up additional scaling from 40 to 60 instead of tapering off as my testing had indicated
Luck = 40/70/99
Item discovery-per-point is again linear all the way to 99, but considering the poor value of the stat most people want it as low as possible, those who invest in Luck have very specific builds in mind. For such builds, AR gains softcap at 40 while bleed buildup scales all the way to 70

Keep in mind there is a fairly good item in this game called the Prisoner's Chain[darksouls3.wiki.fextralife.com] that boosts Vigor, Endurance, and Vitality all by 5 - if you plan to use it adjust your endpoint goals accordingly. Additionally, if you plan to use the Carthus Milkring[darksouls3.wiki.fextralife.com] that makes you invisible while rolling (and can reset any lock-on targeting you) that ring boosts Dexterity by 3.
There are also individual rings giving +5 to one of Strength[darksouls3.wiki.fextralife.com] / Dexterity[darksouls3.wiki.fextralife.com] / Intelligance[darksouls3.wiki.fextralife.com] / -OR- Faith[darksouls3.wiki.fextralife.com], but imo there are better options for your limited ring slots than these.

If you want to see the raw data, it is available here:
https://www.dropbox.com/s/0fqqs5hk3oefy4c/DS3%20Raw%20Data.ods?dl=0

Anyway, I made the rest of this guide mostly as a thought experiment to see which classes got the most out of their starting goodies and how much of a head start they each get. The bonfire measurement is my best estimation for how far into the game a different class would have to go to get the same goodies as the listed class.

Level 9 = 5856 souls
Long Sword = can drop in many places, as early as after the 3rd bonfire
Knight Shield = vendor only item, 1500 souls
Knight Armor Set = vendor only items, 10,000 souls
-----------------------------------------------------------------------------------------------------------------------------
total = 17,356 souls, 3rd bonfire (10th to unlock vendor)

Level 8 = 5063 souls
Sellsword Twinblades = guaranteed item, after 11th bonfire
Wooden Shield = can drop in many places, as early as after the 3rd bonfire
Sellsword Armor Set = guaranteed items, after 11th bonfire
-----------------------------------------------------------------------------------------------------------------------------
total = 5063 souls, 11th bonfire

Level 7 = 4288 souls
Battle Axe = guaranteed item, after 3rd bonfire
Round Shield = vendor only item, 1000 souls
Northern Armor Set = guaranteed items, after 9th bonfire
-----------------------------------------------------------------------------------------------------------------------------
total = 5288 souls, 9th bonfire

Level 9 = 5856 souls
Heal Aid = vendor only item, 500 souls
Spear = can drop in many places, likely only after the 16th bonfire
Talisman = vendor only item, 500 souls
Kite Shield = vendor only item, 4000 souls
Herald Armor Set = guaranteed items, after 14th bonfire
-----------------------------------------------------------------------------------------------------------------------------
total = 10,856 souls, 16th bonfire

Level 5 = 2791 souls
Bandit’s Knife = vendor only item, 1500 souls
Short Bow = vendor only item, 1000
Iron Round Shield = can drop in many places, as early as after the 3rd bonfire
Deserter Armor Set = can drop in many places, as early as after the 3rd bonfire
w/ Thief Mask substitution = vendor only item, 400 souls
30 Wood Arrows = vendor only item, 150 souls
-----------------------------------------------------------------------------------------------------------------------------
total = 5841 souls, 3rd bonfire (10th to unlock vendor)

Level 10 = 6667 souls
Spook = vendor only item, 2000 souls, but interferes with a quest which would reduce it's value to 100 souls
Estoc = vendor only item, 3000 souls
Sorcerer’s Staff = can drop as early as after the 12th bonfire
Target Shield = vendor only item, 2500 souls
Assassin Armor Set = vendor only items, 6000 souls
-----------------------------------------------------------------------------------------------------------------------------
total = 18,267 souls, 12th bonfire

Level 6 = 3531 souls
Soul Arrow = vendor only item, 1000 souls, but interferes with a quest which would reduce it's value to 100 souls
Heavy Soul Arrow = vendor only item, 2000 souls, but interferes with a quest which would reduce it's value to 150 souls
Mail Breaker = guaranteed item, after 4th bonfire
Sorcerer’s Staff = can drop as early as after the 12th bonfire
Leather Shield = vendor only item, 500 souls
Sorcerer Armor Set = guaranteed items, after 12th bonfire
Young Dragon Ring = freely given to other classes
-----------------------------------------------------------------------------------------------------------------------------
total = 4281 souls, 12th bonfire

Level 8 = 5063 souls
Fireball = vendor only item, 1000 souls
Hand Axe = guaranteed item, after 9th bonfire
Pyromancy Flame = freely given to other classes
Caduceus Round Shield = guaranteed item, after 7th bonfire
Pyromancer Armor Set = vendor only items, 2300 souls
Great Swamp Ring = guaranteed item, after 11th bonfire
-----------------------------------------------------------------------------------------------------------------------------
total = 8363 souls, 11th bonfire

Level 7 = 4288 souls
Heal = vendor only item, 1000 souls
Force = vendor only item, 1000 souls
Mace = vendor only item, 600 souls
Cleric’s Sacred Chime = can drop as early as after the 1st bonfire
Blue Wooden Shield = guaranteed item, after 8th bonfire
Cleric Armor Set = guaranteed items, after 8th bonfire
-----------------------------------------------------------------------------------------------------------------------------
total = 6888 souls, 8th bonfire (9th to unlock vendors)

Level 1
Club = guaranteed item, after 3rd bonfire
Plank Shield = guaranteed item, after 7th bonfire
Loincloth = guaranteed item, after 9th bonfire
-----------------------------------------------------------------------------------------------------------------------------
total = 0 souls, 9th bonfire

http://steamproxy.net/sharedfiles/filedetails/?id=684708376
http://steamproxy.net/sharedfiles/filedetails/?id=668170573
http://steamproxy.net/sharedfiles/filedetails/?id=665036873
http://steamproxy.net/sharedfiles/filedetails/?id=665026050

This guide originally started out as the latter part's thought experiment, then I added in the stat breakpoints as an after thought, but since people seemed to like that more I re-arranged the format to feature it more prominently. The data I based the stat breakpoints on is linked above and available for anyone to examine and critique, if you disagree with any of the numbers you can test them in your own game.

My methodology was making a lvl 1 Deprived, using cheat engine (only for the purposes of testing) to give him unlimited souls and the items I needed to test the stats with. STR/DEX numbers were based on bare fists, INT/FTH numbers were based on the pyromancer's flame (I tried getting numbers off staves/chimes/talismans, but the spell buff stat wasn't shown on the level up screen) and the Luck AP was based on a Hollow Club +10 out of convenience since that was the weapon the toon already had. After gathering the test data I needed, I deleted that toon. (btw fighting Iudex Gundyr as a lvl 1 Deprived is kinda fun)

My methodology is not perfect, I cannot make any claims on Luck's influence on bleed/poison since such numbers are no where displayed in game, different weapons have different scaling, and caster foci may indeed perform differently than the numbers I gathered would indicate. I do plan on doing a more in-depth analysis of weapon AP, but that will take a LONG time to complete.

My methodology for that will be to take a toon with 40 in ALL stats and compare the total AP/spell buff of ALL weapons at +10/+5 with ALL possible infusions then repeat the same process again with 60 in ALL stats and analyze the results. This should give a good idea of weapon quality and comparison from a DPH (damage per hit) perspective, but even this will be flawed in that I will not be able to compare DPS without video capture capability and a target dummy that could be smacked around without dying or fighting back as was done by this guy for DS2:
https://www.youtube.com/watch?v=8ar1rnTVzUI
If anyone does such DPS testing themselves, I would certainly host and/or analyze their data if they so wished me to.

UPDATE! I've completed the first pass at 40 all stats, I'm just not sure when I'll get around to the 60 all stats pass since this almost killed me doing it just once >.>