Whether your character can breed rats is a roleplaying question that there aren't really rules for. Even if you've done all the real-world rat-husbandry math, it'll be up to your Dungeon Master to decide how successful your character's efforts are.
For your character to have a rad army of fighting rats that follow your commands is a mechanical question, one that can have an effect on combat, so it's a balance issue.
The DMG describes a magical instrument, the Pipes of the Sewers (page 185), which seem intended to recreate the Pied Piper of Hamelin story but are extremely well-suited to a Rat King character. Merely being attuned to the pipes causes ordinary rats and giant rats to be indifferent to you. (Based on this property, I think there's an assumption that in Generic D&D World, ordinary rats and giant rats are naturally hostile toward adventurers.) Playing the pipes allows you to summon rat swarms and then charm them into obeying your commands.
The pipes are an uncommon magic item, so they're balanced for use by low-level characters. Part of the balancing is wrapped up in the resources you must spend to control rats: You have to expend the pipes' charges to summon swarms of rats, and you must continuously play the pipes (and win a Charisma versus Wisdom check) to make them obey you. Continuously playing the pipes uses your action on each round, so it'll be very difficult to do any fighting yourself while you marshal your army.
This might not be exactly what you had in mind, but it's an avenue for (semi-)reliably convincing rats to fight on your behalf and it's part of the official rules, an advantage that many similarly cool character concepts can't benefit from.
Asking your DM for an opportunity to acquire some Pipes of the Sewers is completely within your rights as a player, and I'd say it's a safer bet than asking your DM for general-purpose Rat King powers—because for the DM, it's just a matter of deciding whether to hook you up with an uncommon magic item.