As stated, the Gameboy Player is needed to run Gameboy games on a TV from a Gamecube- you need both the player (hooks onto the bottom of the Gamecube) and the startup disc (by far the hardest bit to find.) The cable you have is likely just the one used to operate a GBA like a second screen/controller in certain games. It's used, for example, in multiplayer on Zelda Four Sword Adventure & Crystal Chronicles. It can also be used with an E-reader to scan Animal Crossing cards for goodies in that game.
If you want an official means of playing Gameboy games on your TV, the GC Gameboy Player is your best option. The SNES has the Super Gameboy, but it only covers original GB titles. The GC player will run GB, GBC, and GBA. If you're OK with aftermarket options, there's a few floating around- The Innovation adapter from back in the day hooks onto your GBA & runs a composite cable out to your TV. It's hard to find now, you'd probably be better off with the GB player if your already have a Gamecube. Many of the multi-players (your retrons and what-not) have GB slots- anything that does should run the full library, but system build varies so do your homework. There's also the GBA consolizer- that's a kit to turn a GBA into an HDMI-connectible unit. It's currently the only HDMI option (aside from HDMI modding a Gamecube & getting a GB player), but it does involve mod work & sacrificing a GBA to get re-built into the TV variant. You'll need an SNES controller for gameplay, & some places have 'shells' to replace the GBA body entirely so it looks more console-y. The kit's also currently sold out, but you might find people selling it & finished units around the 'net.
