I want to make a character that uses two daggers, but I know that a character using daggers would not be as good as a character using shortswords or almost any other weapon. By "good", I mean approximately the same damage output as other characters. He uses daggers because I thought a character that used daggers would be cool.
How can I make a character that uses only daggers as good as the other characters without homebrew?
Monks gain the Martial Arts feature, one part of which is:
You can roll a d4 in place of the normal damage of your Unarmed Strike or monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table.
And Monk weapons are defined as:
shortswords and any simple melee weapons that don’t have the Two-Handed or heavy property
Since daggers are simple weapons, they qualify as Monk weapons, and can use your martial arts die instead of their normal damage die. From levels 1-4, that doesn't make a difference, because your martial arts die is a d4. Once you hit level 5, however, your Martial Arts die increases to a d6, and your daggers become equivalent to shortswords. At level 11, you Martial Arts die becomes a d8, and your daggers are as powerful as longswords. At level 17, when your Martial Arts die becomes a d10, your dagger is as powerful as a longsword wielded with 2 hands, which is pretty cool.
While the Way of the Kensei Monk (XGtE, p. 34-35) is the Monk subclass that promotes weapon use the most, any subclass of Monk will see their daggers become more powerful with their level. This makes daggers as viable as any other weapon, especially at higher levels.
Yes, the dagger is a measly d4. But the majority of non-heavy weapons have a maximum die of d8 unless you opt to go 2-handed, in which case a d10. But opting for 2 hands generally means that you are opting against a shield for additional AC (although this isn't applicable to all classes due to proficiency.)
The differences in average damage are relatively small between the d4,d6, and d8 dice. You are putting way too much emphasis on the damage die itself and not on the modifiers or other riders that can be added to it. Each increase in die size results in an average increase of 1. So the difference between a d4 and a d6 is an average of 1HP additional damage.
If your concern is over 2-3hp/attack, those numbers really don't matter as much - especially as you rise in level tiers. An extra couple HP is not going to make or break your turn.
The rogue class is where it really showcases how little import the damage die is. Sneak Attack is your primary source of damage, and the difference in damage die there is so minimal that it really isn't a concern compared to the sneak attack damage.
There are various ways to be an effective dual-wielding dagger build. The other answers address some viable approaches; improve the dagger's damage die (by being a monk) or use another source of damage (from a rogue's sneak attack). Below are two more options.
While the shortsword has a larger damage die than the dagger (1d6 vs 1d4), the dagger has the thrown property, with a short range of 20 feet and long range of 60 feet.
Compared to a character who dual-wields shortswords, your character dual-wielding daggers is better defensively, as they can throw their weapons from a distance. Compared to a 2-shortsword character who is limited to melee, throwing daggers lets you target heavy-hitter enemies from a relatively safe distance. And against ranged opponents, you can get up close with your two daggers and attack in melee. So your average damage with daggers is slightly less than with shortswords, but you have more flexibility in your approach.
Otherwise...
The main advantages of a dagger versus a shortsword are (1) everybody can use them, and (2) they can be thrown. A wizard could dual-wield daggers and throw them. If your character isn't leveraging both of these attributes, (i.e. the character is proficient with shortswords and you want to focus on melee combat) then daggers are mechanically a poor choice.
If you only want daggers for aesthetic purposes, then the only important bit is how you describe your weapon. In this case, the smart option would be to dual-wield two shortswords, but describe them as "daggers", or maybe "long daggers" if someone complains. They're approximately the same size (only 1 lb difference), so it's plausible for shortswords to pass as daggers.
There is also an optional Rule in the DMG P270 that enhances the basic bland initiative dice roll with a speed factor.
This option gives a very good initiative bonus do light weapons (like daggers). Maybe this is an optional rule that could be interesting to your group.
The OP's question is asking how they can be more competitive in terms of damage output, while still using 2 daggers. Since using 2 daggers generates the potential for 8 points of damage (+ bonuses) each round, one way to increase this amount would be to multi-class to Warlock.
The Warlock class gives you access to the spell Hex.
Until the spell ends, you deal an extra 1d6 necrotic damage to the target whenever you hit it with an attack.
At 1st level, a Warlock can cast this spell once per short rest and it lasts an hour at a time. This spell increases your potential damage output to 20. At a maximum (before bonuses) of 10 points of damage per hit, that's more raw damage potential than any other one handed weapon in the game.
Granted, because it's a concentration spell, it's not ideal for someone getting hit a lot - but, realistically most mobs should ignore you and your daggers to focus on the giant barbarian with the great axe.
@NautArch brings up an excellent point in the comments below. It's worth noting that Warlock isn't the only class with access to an ability that adds damage dice to each attack. However, in the case of the Ranger's Hunter's Mark and the Paladin's Divine Favor, multi-classing would be less favorable since neither class gets access to their spells until 2nd level.
Another alternative, as mentioned by @RevenantBacon, would be to take the Magic Initiate Feat to gain access to the Hex spell. The downside here is that you'd only be able to cast it once per long rest and it would take up one of your Ability Score Increases if you weren't a Variant Human, but you wouldn't have to sacrifice a level from your main class to do so.
