Most people attach the monitor to a disc, usually cut from wood (MDF or ply), simply because it is easier to cut and cheaper than metal.
On an LCD monitor, the disc can attach to the monitor using the vesa mount on the monitor, remember, you have to insure that the vesa holes are in the center of the screen, if not, the holes will have to be shifted to allow the center of the screen to be in the center of the disc.
The disc is usually supported by bearings or rollers. It's only necessary to have the bearings in the lower left and right "corners" of the disc, however I feel that two in the top "corners" add to the stability of the monitor, especially if the cabinet is to be moved.
The disc must be supported front to back also, to keep the monitor from tipping forward or backward.
Again, bearings or rollers are what most people use.
A metal shaft, in the center of the disc, ran through bearings supported by a block of wood could also be used.
A CRT monitor usually needs two discs, one for the front, with a cutout for the screen, and one in the back, making a "spool" with the monitor sandwiched between the two discs.
These two discs must be attached by wood struts or metal bars, to make sure both discs turn together.
You will need double the bearings for this, because the back disc will rotate also.
You also need to think about how you will attach the monitor bezel, to cover the ugly parts.
I used blocks of wood fastened to the disc, above top and below bottom of the monitor. The blocks extend out to be flush with the edge of the monitor.
I then cut a circle out of poster board, with the. center cut to fit the viewable part of the monitor screen.
This poster cut out attaches to the previous described blocks. This poster board turns with the monitor.
You can then cover all this with another poster, cut to fit the inside of the cabinet, side to side and top to bottom, with a circle cut to expose the screen so that all the screen is exposed when in vertical or horizontal position.
Some people have chosen just to use a smoked glass, but either way, the final cover is the glass (or plexi glass).
I know this is very sketchy, but without knowing exactly what you have, it's hard to say what you need to do, but maybe it will give you an idea how to start.
Next up, choosing a motor and attaching it to turn the disc.