Ok for the meat and potatoes portion.
First things first (im going to assume you know how to take the carb off)
Give the outside a bit of a cleaning before disassembly. I just use a brush and some carb cleaner. Not getting real particular at this point. Get most the crud off, and carry on.
There are 2 sides of the carb. The fuel pump side

And the metering side

Let's start with taking it apart
Remove the screw in the middle of the pump side. Pop the cover off. You'll find a gasket, and the pump valve/ diaphragm membrane.

Set them aside for now. (So you can make sure you have the right stuff from your kit)
Next pop out the screen. This is kinda like a last chance filter of sorts.

Just stick a pick in the center of it and pop it out. This can be trashed. The kits mentioned above only come with one screen.
Next flip the carb around and take the 4 screws holding the primer bulb on, out.

Remove the primer bulb, plastic spacer, metering diaphragm, and gasket.

9 times out of 10 the gasket and diaphragm will be stuck together. Just set them aside for now.
Now remove the needle, lever, and spring. Do not loose the spring! Contrary to popular belief the engine manufacturers spec what spring to use in the carb. The kits may not come with the right spring. They are generic kits for a particular set of carbs. Reuse you original spring.
Theres a wee little screw to take out, that holds the whole assembly in.

Now I don't always do this part, and truth be told it doesn't really need done unless the inside of the carb is really crapped up.
And that is to remove the Welch plug covering the idle passages.

Basically you need to poke or drill a small hole in the center of it, and pry it out.
DO NOT DRILL TOO FAR! I'm talking an 1/8" bit or smaller. Just drill far enough till you can get a pick in the center to pry it out. If you drill too far, DDM sells new carbs.

I mean small hole, and I normally don't even try to drill through it. Just need to get a pick in there. Not held in by much
And here's the idle holes.

Now take your mixture screws out. Turn them in till they lightly bottom out while counting how many turns it takes to bottom out. Write this down so you can reset your carb to where you had it adjusted.

Now this is where my process may differ a bit from yours.
I do tons of carbs a year for small engines. So a long time ago I scored a small ultrasonic cleaner (heated) at a flee market.

It's old as sin. Works perfect though. Toss the carb and all associated parts in it. Fill it with hot water and some
simple green, or whatever you like. (Nothing with strong acid in it. It will etch the aluminum) and go grab a beer, or three...

If you don't have an ultrasonic cleaner. You get to scrub. Carb cleaner works well. Make sure you get it perfectly clean. I will not say I'm a fan of jet cleaner rods. Too much of a chance of messing a calibrated hole up. If you cant squirt some carb cleaner through it to satisfaction, its junk imo. Remember this is a $40.00 part. Not worth it not being right and messing up a $400.00 +/- engine.