This isn't really a house rule, it's a variant rule from the DMG. But I'll write down my explanation of how it works.

You can choose two categories of ability check to be proficient in; one is from your background (and you can choose it freely from the six abilities and come up with a way that it makes sense that a baker is good at charisma or whatever).

The other is from your class:

Pro tip! It's pretty rare to roll an Charisma ability check or Constitution ability check. Those are great for saving throws but not for ability checks. The other four abilities are used more often so I recommend that you chose them. It's up to you of course! They're not completely useless: the downtime rules do use Charisma ability checks and when dashing in a chase you make Constitution checks.

Level two Bard feature. You're just really good at ability checks, you're half proficient in the four you didn't select.

Level seven champion feature. You're half proficient in Strength, Constitution and Dexterity, unless you were already proficient in that ability.

(This ability also boost your jumps but that's unrelated to the whole proficiency issue.)

Level one and six Rogue feature, or level three and ten Bard feature, or optional level 1 Deft Explorer (Tasha) feature. Your proficiency bonus is doubled for one of the abilities you are proficient in.

(Instead of two "skills"; to translate from the old "skill system": two skill expertises can become one ability expertise.)

The Rogue version of expertise also allows you to choose thieves' tools for this, instead of an ability.

As a house rule,

a rogue can choose two of these five:

• Thieves' tools
• Disguise kit
• Forgery kit
• Poisoner's kit
• Any one game set

provided you're already proficient in them.

(This is a house rule.)

Feat. You gain proficiency in one ability and one tool, or in three tools.

(This is a house rule.)

Level three Thief feature. In addition to the normal faster climbing speed and longer jumps, you can add double your proficiency (as if you had expertise) to Strength checks for abseiling climbs, sloping climbs, hard climbs, and very hard climbs. (It doesn't stack; you don't get normal Strength proficiency or expertise on top of that.)

For the other two routes, even with plain Strength you're better than other people at human limit climbs, and superhuman climbs, because normal people automatically fail those climbs unless they have spells. But if you wish, you can select Strength from your background or with the "Skilled" feat in order to make it proficient and then spend your level one or level six expertise in Strength to be even better at those two climbs. That'll also make you a pretty bad-assed grappler

Level one ranger feature & Level three Scout feature.

"Natural Explorer" you make Int & Wis rolls related to your favored terrain (such as foraging, navigating) as if you had expertise.

"Survivalist" same but any nature stuff not just your favored terrain.

"Bonus proficiences" at evel three you get one more ability proficiency.

You get one more ability proficiency.