Before opening the cassette deck or attempting other repairs, make sure it's plugged in properly. If the power cord is attached and it still doesn't turn on, test the outlet by plugging in a lamp or other item to make sure it works properly.
If the deck turns on and spins the tape but produces little or no sound, the problem might be with the deck heads. These heads read the magnetization on cassettes and convert their contents to audio, but they can become damaged over time. Use a tape designed to clean cassette heads to see if that fixes the problem, or reach into or open the cassette deck, and swab the heads with cotton and denatured alcohol or cassette head cleaning fluid.
Cassette heads can also become magnetized over time, which can cause them to fail to work properly and even damage tapes. Electronic stores sell various types of demagnetizers, also called degaussers; follow the instructions, and test the deck.
Because cassette decks are mechanical devices, they contain a number of moving parts that can fail to work properly over time. Opening the cassette deck and seeing what motors turn can help determine the problem, but avoid touching anything in the deck while it's plugged in to avoid shocks. Capacitors and other components can carry charges even after the device is turned off. Decks typically use manufacturer-specific internal components, so it might be best to hire an expert if the deck has a mechanical problem.