So it looks like my "use every winning proficiency" character is ready to go!
Artificer, level three.
Yep, just roll up any artificer, and advance it to 3rd level. You now have all the tools, hooray.
Okay, okay, I'll tone down the snark. Having the tools doesn't make you proficient with them. And Artificers are lame, besides. So let's give this an honest attempt...I'm now going to make a playable, interesting, serious character with these five Tool proficiencies.
I always create a custom background, using the others as a guide. So let's make a background called "Alchemist's Apprentice," and pick up Cook's Utensils and Alchemist's Supplies. I'm a plucky teenager named Boxley, and I'm an unpaid intern (and housekeeper) for a popular alchemist in town.
This isn't going to matter much, I'm afraid...no character race or subrace offers the remaining tools as a proficiency for free. If my DM allows it, I could ask to swap the two Skill proficiencies for Tools instead, but as-written? No race options give these tool proficiencies by default, so it won't matter. Honestly, tool proficiencies are awful. And useless.
You don't need to be proficient with a Disguise Kit to attempt to disguise yourself. You don't need proficiency in Cooking Tools to make yourself something to eat. Proficiency will let you add your proficiency bonus to the check, sure, but by the time your PB is high enough to matter, you've found better and faster ways to disguise and feed yourself. All proficiencies are like this, but Tools are the worst offenders IMO because they imply that only skilled people can use them. And that's just not true, not in D&D and not in real life. Seriously, 9 out of 10 people don't know the difference between a boning knife and a paring knife, but they still manage to cook breakfast every morning without poisoning themselves. Ugh. Anyway...
The best you can do is choose the Variant Human, and give yourself the Skill Expert feat at 1st level. Then promptly waste it, by using it to pick up some tool proficiencies. For this exercise, I'll grab Disguise Kit and Cartographer's Kit. These will never matter.
Class and Subclass:
The Druid class will give me proficiency with the Herbalism Kit, which is arguably the only "useful" Tool on the list (it allows me to make my own healing potions). So let's say that my Master is chronically ill, and so the first thing he taught me was how to prepare his medication. The druid class also has other skill proficiencies and magic that would make this "apprentice" character a little more fleshed out: I grew up in a druid enclave, but ran away to the Big City to find my own path and ended up here.
For the subclass, I would choose Circle of the Stars just to get the star map. Now I have a story-based reason for the Cartography Tools proficiency. Which is good, because there certainly isn't any other reason to select it.
I now present:
Boxley, Apprentice to M.Sorill, Renowned Alchemist
Race: Human (Variant)
Class: Druid (Circle of Stars)
Background: Custom, "Alchemist's Apprentice"
- - - - -
Boxley was born and raised in the wilderness of the West Forest, and was brought up in the ways of the Druidic Enclave. But unlike most of his friends and family, he was born with a wanderlust...he dreamed of someday leaving the forest village and visiting the large, glittery cities that he could see down in the valley. And so it came to pass that one fateful morning, Boxley set his heart on finding his fortune in the big city. He waited until his parents were both asleep...then quietly retrieved his backpack from its hiding place, grabbed his walking stick, and quietly set out for the Great Unknown.
"He'll be back," his father said quietly. "He will find the thing he's looking for, and it will break his heart, and then he will return." His voice carries the weight of sorrow, but also pride.
"He gets that from your side of the family, you know," Boxley's mother replied, watching their son's best attempt at sneaking past the window and shaking her head. She squints to get a better look, and furrows her brow. "Why did he take our spatula?"
- - - - -
Two years later, and Boxley is up to his elbows in soapy water. This is the fifth set of vials he has to wash, and by the time he's finished with these, there will barely be enough time to finish dinner. Master Sorill is very particular about his dinner, and he expects it ready at exactly six o'clock...wait. "That's strange," Boxley thought to himself. "Master Sorill hasn't returned yet. He should have been back an hour ago. I wonder what's keeping him?"
- - - - -
But as fate would have it, Master Sorill would never return. Now, with only his master's journals to guide him, the young M.Boxley sets out on an adventure to find his lost master.