dnd 5e – How can I win competition in medium armor like a rogue?

All the methods to obtain a medium armor competition are based on optional rules, which means that you must eliminate them with your DM.

Multiclases

One method would be the multiclass, as you say, which is explained in the Player's manual in chapter 6. The chart on page 164 shows what you earn by making multiclases in each specific class. Many of them can grant an average armor competition, but keep in mind that you must meet the appropriate minimum statistics requirements that are listed on the previous page.

Feat

The simplest method would be to take the "Moderately Armored" feat (p.168) when you climb up to level 4, instead of increasing your usual skill score.

But are you sure you really want to wear a medium armor?

As a rogue, you almost certainly have a very high Dexterity, and in that case, you should wear light armor. You are for whom it is made. While a rogue multiclass character may wear a medium armor, it is usually because they do not focus on Dexterity for some reason, as they are mostly a fighter with only a bit of furtive attack mixed. Armor for a very good reason.

First of all, keep in mind that the Scale and Half Plate armor gives you a disadvantage in Stealth, which you really do not want, like a rogue.

Second, the medium armor only allows you to add up to +2 of your skill bonus to your CA. As a result, with a Dex of 14 or more, the average armor ACs go from 14 (hide) to 17 (half a plate), and most likely end in 16 unless you are willing to drop a moderately large amount of money on half a plate (which, again, makes it hard to sneak away).

With a studded leather armor, which is cheaper than any medium armor, except the skin, its AC is a 12x Dex modifier, with no dexterity limit. If you have an initial Dexterity of 16 or 17 (+3), then your AC with studded leather would be 15. Add +2 to your Dexterity with your 4th level Ability Score Increase, and you will get an AC of 16, which is like Good as any other medium armor that is not half a plate. At level 8, you can increase your Dexterity to 20, at which point your old leather armor with studs gives you AC 17, which is as good as anything else you can get with a medium armor.

If you started with Dexterity 18, you will reach that maximum CA of 17 even faster.