[RPG] How to a human be viable in an Underdark-focused game

Your PC can invent a new technique on his own

Taken from wikipedia's entry on the Southern Praying Mantis kung fu style:

Praying Mantis

The association of the term "Praying Mantis" with the style is also controversial. Each branch of the style offers a different explanation.

The traditions of the Chow Gar and Kwong Sai Jook Lum branches each maintain that their respective founders Chow Ah-Nam and Som Dot created their styles after witnessing a praying mantis fight and defeat a bird. Such inspiration is a recurring motif in the Chinese martial arts and can be found in the legends of Northern Praying Mantis, both White Crane styles, T'ai chi ch'uan, and Wing Chun.

If your rogue is observant enough - he may become inspired by some natural phenomenon, gaining insights about efficient and effective motions useful for offense, defense and maneuvering.

He doesn't need to be an acetic philosopher (though he may grow to become one as he advances) - he just needs the conviction and self-discipline to push him to go through the physical training required to hone his technique.

If your group's play style allows placing more emphasis on narrative at the cost of poorer combat optimization, you can even select to make the transition of your PC's reliance on his new martial skills gradual - continue wearing armor, combine armed attacks with unarmed strikes (not necessarily in the same round, though), etc. But, eventually your PC will start feeling that his armor is limiting him, preventing him from escaping blows that using his technique he can now avoid - then he may consider giving up on wearing armor. Same goes for unarmed strikes - have him hold on to his dagger, but occasionally use a flurry of blows⁽¹⁾ using a fist and an elbow of his other, empty hand - and let him realize that his hands have become faster and more accurate and than any weapon he is used to. This can lead him to relinquish the use of melee weapons, to use an interchanging combination of weapon and unarmed strikes or perhaps even seek "better" weapons which are more compatible with his developing abilities.

Bottom line - just because the "metagame concept" says your PC can now do something, doesn't automatically means he is aware of that and trusts himself enough to give up his old ways completely and immediately.

Or, he can find written records of a technique

Taken from wikipedia's entry on the Northern Praying Mantis kung fu style:

Origins

There are many legends surrounding the creation of Northern Praying Mantis boxing. One legend attributes the creation of Mantis fist to the Song Dynasty when Abbot Fu Ju [...] supposedly invited Wang Lang and seventeen other masters to come and improve the martial arts of Shaolin.
The Abbot recorded all of the techniques in a manual called the Mishou ("Secret Hands")
[...]
This manual supposedly disappeared until the Qianlong reign era,
[...]
The manual records Wang Lang "absorbed and equalized all previous techniques" learned from the 17 other masters.

Aside from self mastering a technique, your rogue may find a documentation of a certain technique - this can come instead of having him invent it from scratch, or as a latter supplement giving him more inspiration and advanced abilities.

At either case, you can probably work the narrative details with your GM so they won't clash terribly with the campaign settings (assuming that having a monk in it doesn't do that in the first place).

(1) Or whatever equivalent term used in 5e for that...

No, you cannot stack proficiencies.

You're either proficient with a skill/item/weapon and get to apply your proficiency bonus, or you're not; you can't be doubly or triply proficient in something. Some class features and special abilities may allow you to double your proficiency bonus in some situations, but these are always explicitly spelled out. As the PHB states on page 12, under the "Proficiency Bonus" heading:

Your proficiency bonus can't be added to a single die roll or other number more than once. Occasionally, your proficiency bonus might be modified (doubled or halved, for example) before you apply it. If a circumstance suggests that your proficiency bonus applies more than once to the same roll or that it should be multiplied more than once, you nevertheless add it only once, multiply it only once, and halve it only once.

And this is reiterated in further detail in the "Proficiency Bonus" section on page 173:

Your proficiency bonus can't be added to a single die roll or other number more than once. For example, if two different rules say you can add your proficiency bonus to a Wisdom saving throw, you nevertheless add the bonus only once when you make the save.

Occasionally, your proficiency bonus might be multiplied or divided (doubled or halved, for example) before you apply it. For example, the rogue's Expertise feature doubles the proficiency bonus for certain ability checks. If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you still add it only once and multiply or divide it only once.

If you would be granted the same tool/skill proficiency by your background and your class during character generation, you get to just choose a different proficiency of the same type.

Extra: However, as mattdm's answer points out, Xanathar's Guide to Everything introduces a new feat, Prodigy, which amongst other things allows you to choose one skill you have proficiency in and double your proficiency bonus when using that skill - so as a variant human taking a feat at 1st level, you could improve your bonus to one skill in this way.