How do I tie a closet I'm building into the floor and ceiling?

Here is a basic instruction of how to build a closet. I hope this helps you! https://www.wikihow.com/Build-a-Closet

There should be no reason to cut away flooring, unless you're referring to the baseboard, so the closet sides are flush with the wall. Floors are not cut away for closets.

You don't need to cut into floor or ceiling, just finish the top and bottom with quarter round, or whatever edging you like the best. That should tie it into both the floor and the ceiling.

To tie the top and bottom into the floor and ceiling you need to use a sole plate (bottom) and a top plate. It is important that they are exactly aligned with each other. To do this you can use an old fashioned ‘plumb bob’, or a laser level. Then you nail into the studs on top and the floor on the bottom. To do the sole plate you lay one solid 2x4 straight across (do not cut away for the door yet. This is so that the walls on either side of the door are perfectly aligned with one another. Once the sole plate is installed and nailed down you can cut out the piece where the door will be. You can use a framing square to ensure both corners are square. Lastly, do the framing and drywall, or whatever you are going to use for the outside walls.