Let's keep simplifying until we're a rules-lite hipster clone, and while we're at it keep compatibility with other characters and published adventures.
First off, we need to know what exactly can be condensed and simplified. Class abilities and spells do not seem to have a convenient way to be condensed, but there's enough space on the card to leave them as-is. In the Dungeon Master's Guide there's an option to replace skills and tool proficiencies. While they're not a huge chunk of your character, they still take up quite a bit of space. The Background Proficiency variant allows you to instead roll skill and tool checks with your background instead, so let's go with that option.
Racial traits can take up quite a lot of space as well. Instead let's just toss out everything but Ability Score Increases and give characters a Racial Proficiency instead. It works a little like Cliches in Risus: Elves get a proficiency bonus for doing elfy stuff, dwarves get a bonus for doing dwarfy things, so on and so forth. It does make the races boring as a lot of the more flavorful abilities get axed, but we're going for as light as possible here! Humans, meanwhile, get to pick a second background proficiency because Racial Proficiency: Human is just plain silly.
Now that skills and racial features have been axed and replaced accordingly, a look at equipment proficiencies is in order. Weapon proficiencies are split into Simple and Martial, but some classes have exceptions. Just remove the exceptions and leave it at that. This does mean the wizard and the sorcerer get no weapon training at all, so if you want to allow them simple weapons that's okay too. Armor, meanwhile, is just Light, Medium, and Heavy. There is no class that gets shields separately from medium armor so it's okay to lump shields in with those.
For tracking encumbrance, it might just be easier to say that you can carry a number of items equal to your Strength score, barring things that are worn/held (weapons, clothing, armor, backpack, etc.). Objects that come as a set or have a very low weight should count as one item when they're a full set or more than enough to be at a meaningful weight (10 or so). A good guideline is, if it can fit in the palm of your hand it does not count against encumbrance. If it can be held in one or two hands, it should count against encumbrance.
With a good chunk of the character sheet now gone and condensed into a few proficiencies, a first level character could fit onto a 3x5 card with little difficulty... If one is playing a martial class. Spellcasters need another index card to record spell slots and spells. Or just use the back, but you need that space to show off your sweet character doodles.
|If the back of your character sheet doesn't look like this something is probably wrong with you.|
A sample character would be as follows:
Ivan Hammerfall (Fighter 1) St 15 Dx 10 Cn 16 In 13 Wi 13 Ch 8
HP: 13/13; HD: 1 (1d10); AC: 17; Saves: St, Cn; Prof Bonus: +2
Background Prof: Soldier, Dwarf; Arm/Wep Prof: All
Fighting Style: Defense (+1 AC with armor)
Second Wind (regain 1d10+level HP, 1/short rest)
Gear: Chainmail, longsword (1d8/d10), light crossbow (1d8/ammo 20)
Backpack: Insignia, scrap of banner, deck of cards, common clothes, bedroll, mess kit, tinderbox, 10 torches, 10 days rations, waterskin, 50 ft. rope
Money: 10 gp