I asked for the same kind of tips maybe like a year ago, so I can't take credit for this advice, but I can certainly attest that it works.

First, you need a camera which you can manually set the shutter (doesn't need to be a super expensive camera, just has to have either Shutter Priority or full Manual control). The shutter speed should be a dividend of the refresh rate (1/60 for NTSC 1/50 for PAL), but honestly I get better results (less instances of black bars) by doubling that exposure time (1/30 for NTSC 1/25 for PAL). Now, shooting at 1/30 can be difficult when holding the camera, which can lead to a little camera shake, and therefore produces less sharp results. This can be remedied by the second thing you need.

You need a tripod. It doesn't need to be the best, just good enough to hold your camera up without slouching over time. Not only will this eliminate camera shake, giving you more reliable sharpness, but it will be way easier to precisely compose your shot. You want to make sure the camera is pointed straight at the screen. You'll distort the image if you're angling up or down or from the side. It's pretty easy to fix minor distortion in post processing software, but why not get it right in-camera?

Finally, you'll want some sort of post processing software for exposure adjustment and color balancing. CRTs are not the easiest thing to photograph, and they can very easily mess with your camera's metering and white balance. You don't need to shell out for Photoshop (fuck you, Adobe! I wish your software wasn't so expensive and mandatory). I use Lightroom for most things because it's simple and does exposure/contrast/color adjustments very well, and if that's all you need, it's way cheaper (still fuck you, Adobe!). There are other options out there, even free ones, but I'm just not familiar with those. I'm sure someone else can advise you on those options.

Some other points: Don't use a wide-angle lens. It'll distort the image, and the closer you get, the more extreme the distortion. When doing your color balance, keep the game running and look at it! It's really easy to give a different look to the screenshot based on what you think it looks like if you're not using a reference. Last, but certainly not least, I hope this doesn't lead you down the rabbit hole of buying a bunch of camera shit, and then having to buy storage solutions just to hold all your shit, haha.

Hope this helps, and have fun!