I'm writing a story where people suddenly gain a 1st Level D&D 5e Character with Background, Class, and Ability Scores. Each person also gains the racial abilities of whatever race they choose, without the Age and Size, so they still remain Human.

These select individuals have to protect the world from typical D&D monsters that start appearing all over the world.

Along with all this, each person is capable of seeing their abilities in their own personal HUD (heads-up display), just as if it was a video game.

The problem I'm running into is that a normal (average) D&D combat encounter only last 4 to 5 rounds, which is only 24 to 30 seconds per fight.

My solution is possibly to increase the PC levels to 100. At levels 5,10, 15, 20, 25, etc. will be equal to levels 2, 3, 4, 5, 6, etc. Using this setup, the PC's will get more HP, but still have all other abilities and attacks as normal.

The monsters will also get a boost to their HP ×5. The monsters abilities and attacks will also be the same as normal.

I was hoping it would just make combat encounters long, i.e. making the fights longer and more cinematic. I'm also hoping it doesn't allow characters to one-shot monsters with one hit.

I think that you may be overthinking the issue with combat length. Here are some easy ways to overcome it cinematically:

• Think of those actual combat rounds as the ones that actually have impact on the story.  You can fill the space between with those 'rounds" where weapons hit shields, spells smash into the cover someone ducks behind etc.
• If you have ever read RS Salvatore, I would say his combats are typically short in duration.  He just uses great description to give each strike and manuever and spell flavor and context that builds the cinematic feel,

Just a couple thoughts.

To build on Grebeir’s great suggestions you can describe the action however you want using a D&D round as a basis. But also consider how long an actual combat takes to play out in game. It doesn’t happen in 25-30 seconds. Describing it in a story can likewise take longer than the actual combat itself.

Is there a reason you want to strictly adhere to a D&D combat round?

Consider a party of five against a group of 8 monsters. Each player/monster gets 6 seconds and although in game they all happen simultaneously describing each ones actions can take a sizable chunk of real estate on the page. Do you really want to spend so many pages on a combat? Can you keep the readers interest if it drags on turn after turn, round after round?

Start with a standard 3-4 round combat, write it out with vivid descriptions, like Grebeir said, and see what you end up with. It might be better than you have anticipated. 

You could always scrap the 'one round = 6 seconds' thing. I think that is a far better idea than bloating combat length with massive HP pools. 

You could then ignore the six second thing and rework spells and abilities. A spell that lasts 'one minute' you change to be a more nebulous '10 rounds' when in combat. A 10 minute spell is 100 rounds in combat, or 10 minutes out of combat. 

The duration of spells and abilities are really the only thing that the 'six seconds per round' thing are critical for. If you just change those to last a set number of rounds in combat, you're good. 

Sounds like a cool premise, and I agree with others before me that getting bogged down in the exactitude of the combat round length is probably something worth dropping. You can't be doing it to maintain a perfect analogue with dnd, or you wouldn't be suggesting changing the HP pools and levels!

One thing to consider is the perception of time. There is an excellent low fantasy book series called The First Law by Joe Abercrombie, and that features a sword fight between two masters. The fight lasts 2-3 pages, blow by blow, as their masterful skills are put to the test, from the perspective of one of the duellists. Afterwards, the perspective changes to a bystander, who comments (in their mind) about how it happened too fast to keep track of, lasting no more than a few seconds. This was an excellent approach by the author to emphasize the skill - so much stuff happened, and then it is revealed just how quickly it happened.

I would ofcus on making it an enjoyable book to read. No-one will be concerned with exactly how long the combat lasts, how long there is between each blow or movement. Your focus needs to be on the feel of the fight. If it's well structured and the people are focussed, give it blow by blow. If the perspective is that of someone who's scared, or dazed, or not concentrating, or the fight is chaotic, make it clear that you're not describing it in perfect detail. a sword thudding into a tentacle. Dwrmun the dwarf shouting. Something spraying in her eyes, an inhuman shriek following. A sudden bright light, the floor coming up to meet her. To me, that is far easier to follow and enjoy than a blow by blow as someone attacks the tentacle, then dwrmun attacks it and cuts it off, then the thrashing stump catches her in the head and knocks her to the floor. The implication remains - and anything too cryptic can be cleared up later ("ted the bard saw the fighting intensify - saw Dwrmun cut off a tentacle, and saw the flailing stump catch Cecilia and knock her to the floor, and he sang all the stronger to help her get up"). The more chaotic a fight, the more you spread out the information and let the reader piece it together by the end. 

Honestly, I've never read a book where it breaks down the combat into 6-second intervals, or even concerns itself with how long it takes, exactly. It instead focuses on the consequence of the time taken. "By the time the fight was done, the beast had already climbed halfway up the eifell tower and was pulling its collection of giant marbles from a large bag and throwing them into traffic", rather than "the fight lasted 36 seconds, during which time...". a group of 5 people fighting 5 monsters for 4 rounds is, if you broke it down entirely, 40 actions to describe. You run the risk of making combat the least interesting part of the book!