What a fantastic idea! I’m surprised it hasn’t been done like this before. Last year, a site called Chesscademy launched to teach you chess for free. Clearly a play on Codecademy, but brilliant nonetheless.
The course covers beginning, intermediate and advanced chess skills. It’s taught by Andrew Ng, a 7-time US National Chess Champion. I find the instruction clear, and easy to follow (at least at the beginning stages).
The interface is nice and intuitive, and it all works really well together. Videos that are not too long are followed by a short quiz, proving that you learned the lesson taught. The quiz interface naturally contains a chess tool, so that you can demonstrate a move by dragging a piece on the screen. And you usually also have to answer some multiple-choice questions as well.
The training is free. If you have an interest in chess, I encourage you to check it out.