GoKawiilWith the right variant, playing GameCube games on an LCD or OLED TV is as simple as buying an adapter and connecting the two devices over HDMI. While you can find plenty of cheap HDMI adapters online, I recommend spending a bit more to buy the Retro-Bit Prism HD Adapter. Retro-Bit makes great stuff, and its GameCube adapter is no exception. Most GameCube HDMI adapters rely on open-source software called GC Video. The Retro-Bit comes with a USB-C port to make updating that software easy; no need to take apart the device.
Another option is the Carby from Insurrection Industries or, for a more affordable alternative, consider Bitfunx's offering. Just note, depending on the dongle you buy, you may also need to get a separate HDMI cable if don't have a spare.
Once you have an adapter, all you need to do is plug it into your GameCube's Digital AV Out port, and then connect an HDMI cable from the adapter to you TV. All three of the models recommended above come with a remote that allows you to bring up GC Video's on-screen menu to adjust settings like output resolution.
Now this is where things get a bit more complicated. To get the best image quality from your GameCube, you'll want to consult this webpage to find out if you own any games that support 480p output. If you find one that does, pop it your console first and hold down the B button on your controller as the game boots up. You will see a dialogue that asks "Enable progressive scan display?" Select Yes. Provided the game supports this feature, progressive scan will give you the best possible image quality from the old console. Some games also support native widescreen output. Typically, you can enable this from the settings menu of a compatible game.