Before going into specifics, some general tips:
STAY ON THE PLATFORM!!!!!! Very simple, but easy to forget. If you get knocked off, use the Whiplash and snag some heavy enemies, or use an explosive to knock yourself up.

Ration the Railcannon. Often times I forget I just used the Railcannon and waste time when I could be fragging. Also, make your Railcannon shots count!!

PARRY THIS YOU F***ING CASUAL!! Using parries are a surefire way to get topped off on health from a distance. Use this to your advantage.

Quick swapping! It's super effective at maximizing damage and freshness bonuses. Constantly pressing 'E' will swap between equipped mods. Each mod can be used in conjunction with each other, so try out all of them on your own time! Here are a few well known ones that I enjoy...

SHOT-SWAPPING: Quickly switching between the Pump and Core Eject shotguns removes the cooldown between each shot, so you can maximize your damage. If the firing audio cuts out, you're doing it right.

SHOT-PUNCHING: Parrying your shotgun pellets. Press 'F' with the Feedbacker (blue arm) about 0.000000001 milliseconds after you fire to give your shotgun explosive rounds and do a crap ton of damage. It could take some getting used to, but once you get into the hang of it, it's more addicting than heroin.

CANNON-LASHING: This is where you fire the S.R.S Rocket Launcher's Cannonball and let it hit off an enemy, stopping its momentum, then Whiplashing the Cannonball (pressing 'R' once you beat V2 a second time) and bringing it close to you. Then you can parry the Cannonball. Effective at taking out Cerberi

Make sure to experiment and try all of the various combos out!
(I know I left out Railcoining, I'll get to that in a bit)

Enemies come in all shapes and sizes in Ultrakill, from sexy robots with skin tight latex to giant rocks that fire laser beams. Knowing your enemy is good, but knowing what they can do to you is even better. First, observe a new wave every time it spawns in. If there are any Idols, take those out IMMEDIATELY!!!! From there, focus on the biggest thing you can see, and work your way down.

Part 2.5: Leave the little guys alone...for now.
Letting fodder demons survive for a while can be a worth-wile strategy. For example, if you're taking heavy fire from a Mindflayer, dash over to a local Stray and mug him for his blood, giving you a quick refill before you go back to fight that Mindflayer. Leaving fodder demons alive can also let you top off your health at the very end of a wave, making sure you're fully refreshed and ready to fight the four Swordsmachines and twelve Malicious Faces next wave.

DO NOT NEGLECT THE KNUCKLEBUSTER!!! This thing can be used to knock Stalkers up and away, quickly kill a fodder demon without wasting freshness, as well as damage a group of enemies with the shockwave.

As well as this, the Whiplash is a very important tool for zipping around the map. Just don't forget you're pretty vulnerable while doing this. The Whiplash can also do some cool things, like grappling onto Cannonballs, Core Ejects, Frozen Rockets, and even coins! Flicking a coin or two into the air, then hitting one with the Whiplash with shoot it up a bit, giving you more time to hit Chain-Richoshots.

The Richoshot Coin is such a versatile weapon, and when used with the Railcannon's Electric shot, can easily decimate anything in it's path.
Chaining coins is efficient, but risky, as it forces you to use more coins, which means more down time. I recommend throwing one or two coins up at a time so that you can maximize damage if you get into a pinch.
Railcoining is difficult to pull off, but oh so satisfying. Hitting the coins with the Electric shot can eliminate almost any threat. Just make sure you don't miss.
I recommend the default Richoshot, not the Slab, so that you can split-coin, and hit two demons with one bullet.

Movement is a core part of Ultrakill, and the Cybergrind is no exception. Dashing is helpful, but I find Slam Storage to be the most effective. Slamstore, then launch a few coins to maximize the down time. Sliding isn't all that great in the Cybergrind, but it has its moments. Jumping is huge, obviously, but so is GroundSlamming. If you're launched too high up, GroundSlam to return quickly and efficiently, and maybe take out some poor sap on the way down. Trust me, GroundSlamming has saved my metal a few times more than a lot.

Remember: you're pretty squishy and can die easily...but so can they. Maximizing your damage is a surefire way to stay alive in the heat of things. I hope this guide helped you at least a little bit. If not, well...I dunno go watch Family Guy or something