I’m playing a blind (but with 60ft blind-sight) half-elf bard with animal handling and animal friendship in a D&D 5e game.
I'm interested in capturing two wild rats and breeding them into any army that scouts, attacks and defends on command.
Is this doable? I looked into litter sizes, gender ratios and gestation periods. I assume 600 rats in a year’s time.
Judging from this question, there are no rules for training animals, so right off the bat you are entering homebrew territory.
If your DM is willing, then maybe you could train rats to help you. You can use spells such as Beast Bond and Speak With Animals to try and communicate with them. Talk to them, bribe them, and train them using these spells. (Animal Friendship may not help much, other than to let you try to persuade it, which wouldn't work unless you are speaking with it.)
Disclaimer: A rat's INT score is 2. Training it may be very difficult.
All in all, it is up to your DM whether or not you can do this.
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.
The first thing to come to my mind when I read your post was the Swarmkeeper Ranger subclass (Tasha's Cauldron of Everything), but later I remembered the rules for sidekicks (same manual). Maybe you could talk with your DM to use the swarm as a sidekick with the ability to drop single rats to explore. The only catch to this approach (that I find) is that sidekick rules say:
A sidekick can be any kind of creature with a stat block in the Monster Manual or another D&D book, but the challenge rating must be 1/2 or lower. You take the stat block and add to it, as explained in the "Gaining a Sidekick Class" section.
And it is not completely specified that swarms are creatures, I'll leave you this thread on the subject: How are swarms considered in terms of number of creatures?
