So me and my dm are arguing about if it is possible to make a Hexblade warlock with plate armor using Tavern Brawler. I argue it should work since Tavern Brawler allows you to be proficient in anything and turn it into an improvised weapon, including plate armor and other random objects. Comboing this with Hexblade which allows you make a weapon you are proficient with into your hex weapon, I thought it would be cool to have a full-plated eldritch knight without a weapon and use his fists to smite everyone. Does this make sense or is it not legal?

Most dms don't consider improvised weapons as actual weapons from the weapons table in the equipment section. Bones can be improvised weapons and function as clubs, but they aren't weapons people train with to acquire proficiency with in most campaigns.

There are four main categories of weapons: Simple Weapons, Martial Weapons, Improvised Weapons, and Natural Weapons.

Simple and Martial Weapons are what are defined in weapons tables in the PHB and other books.

Improvised Weapons are defined in the PHB as "An improvised weapon includes any object you can wield in one or two hands ..." - https://www.dndbeyond.com/sources/basic-rules/equipment#ImprovisedWeapons

Natural Weapons are specific body parts that are particularly well suited to dealing damage. The Monster Manual provides the closest thing we get to a formal definition saying "These can be spell attacks or weapon attacks, where the “weapon” might be a manufactured item or a natural weapon, such as a claw or tail spike." - https://www.dndbeyond.com/sources/mm/introduction#MeleeandRangedAttacks

What about Unarmed Strikes? Well they are defined in the PHB as a Melee Weapon Attack made without using a weapon. - https://www.dndbeyond.com/sources/basic-rules/combat#MeleeAttacks

So unfortunately according to RAW a donned suit of armor is not an Improvised Weapon, as it is not an object wielded in one or two hands, and Unarmed Strikes are defined by their very lack of using a weapon.

Personally I think this could make for some fun builds, and I don't see anything game breaking about it. But you should talk to your DM about bending the rules as it is not allowed by them as it stands.

If you want to use plate armor to hit somebody with as an improvised weapon then have at it. But Tavern Brawler will not give you the armor proficiency to wear. Unless I am mistaking what you are saying. 

If you want to wear heavy armor you will need the Heavily Armored feat or multiclass to get it.