ClassiCube

First, go to this link: https://web.archive.org/web/20150925124754/https://minecraft.net/classic/play After that, head to your Classic Sharp Launcher and open it. Next, sign in and head into either Single Player or Multiplayer! Then, close the launcher with the server list, and keep the game on. Last but not least, Put the opened launcher in the middle of the website.

It looks just like this: http://prntscr.com/ar4b12

In the screenshot, look directly in the middle, you will see that the black border blends with the website wallpaper! And that's how you play Minecraft Classic on the website.