So I recently had the idea to buff monk after I and many of my usual group realized how underwhelming the base monk is, so here is my pitch to make monk slightly better yet still unique.

Starting at level 2, the monk gains a new tactic... Chain Martial Arts. Every time your monk character lands a melee attack, you gain 1 extra attack up to your proficiency bonus. This feature may used in addition to the Extra Attack feature. Once the max amount of Chain Attacks have been reached, you cannot use this feature until you finish a short or long rest. 

Clarifications: I know the upper text is written crudely, but my attempt to replicate the style of writing found in the books is subpar. So the idea behind this feature is that monks do little damage for 1 strike, but they will be able to do more attacks to even the playing field. So lets say a Level 5 Monk (Ignoring Subclass), uses this feature. They have +3 proficiency bonus, can make 2 attacks via Extra attack, and spend 1 ki for Flurry of Blows. This means if they connect every attack on their turn, they can expend all of their resources to make 2 regular attacks, 2 attacks (FoB), and 3 Chain attacks for a total of 7 attacks (at least 5 are melee from Chain Attacks and Flurry of Blows).

Is this Busted? MATH WARNING! The Creating a monster chart in the DMG states that average CR 5 creature should had an AC of 15 and 131-145 HP. Now if the monk in question has an 18 in DEX and is wielding a quaterstaff with both hands (unarmed strikes can be kicks, knees, elbows, and headbutts), they have a +7 to hit with the quaterstaff and unarmed strikes aka 65% chance to hit our CR 5 creature. So the equation well use for the expected damage will be %(To miss) x 0(damage) + %(To hit) x (Expected damage of dice). This equation can be found and explain here (https://tabletopbuilds.com/a-primer-on-quantitative-evaluation-in-dd/). 

So the expected damage for 1 round is (((0.35 x 0) + 0.65(4.5)) + 4) + (((0.35 x 0) + 0.65(4.5)) + 4) = 13.85 damage from attack action w/quaterstaff both hands (Action).

(((0.35 x 0) + 0.65(3.5)) + 4) + (((0.35 x 0) + 0.65(3.5)) + 4) = 12.55 damage from flurry of blows (Bonus Action)

So far this is standard monk, but I'm going to have to wing to math here for this feature here on.                                                     (((0.35 x 0) + ((0.65 x 0.65 x 0.65)10.5)) + 12) = 14.88 damage from Chain Attacks (Free Action).

All together we can expect around 41.283 damage for 1 round per short rest. 

I know that sounds like a ton, which it is but I think this is largely balanced for 3 reasons. 

1) It forces you into melee, which as a monk is terrible due to the fact you have average HP and typically low AC. This feature gives a risk/reward mechanic that I believe monk currently lacks. Deal Massive burst damage, but risk getting shmacked yourself. 

2) Other Melee Builds can dish out similar damage. Level 5 Fighter, 18 STR, No feats or subclass, using a Great Sword will deal around 17.1 damage per round with Extra Attack = (((0.35 x 0) + 0.65(7)) + 4) + (((0.35 x 0) + 0.65(7)) + 4). This is without expending anything (Our Monk's DPR is around 5.85), but if we want just want 1 round then we can double our DPR once using action surge to do it again for 34.2 damage. That is still around 10 damage below our monk, but fighter is much more suited for head on combat with higher AC and HP, along with an one use self heal. 

3) It makes monks and rogues even more different. Vanilla Rogues do massive damage in 1 hit using sneak attack, but vanilla monk does mediocre/ bad damage with many strikes. With this homebrew it makes monk do incredible 1 round damage using many strikes, and I think it will give more flavor into parties.

TL;DR: Make many hits, rack up damage, risk getting shmacked, kinda balanced.

What do Y'all think? Maybe make this feature at a later level? Make it incompatible with Extra attack within that turn? I am open to criticism.

 (Our Monk's DPR is around 5.85)

Where are you getting that from? Using your own calculations, a monk with a quarterstaff using the Martial Arts bonus action would do (0.65(8.5) + (0.65(8.5) + (0.65(7.5)) = 15.925. Less than 17.1 but way more than 5.85.