The easiest and simplest way to connect up your gaming chair is via the audio out sockets on your TV/monitor.
Using the relevant audio cables that accompanied your chair connect to the audio out sockets on your monitor and the audio in socket(s) on your chair.
Be aware that a headphone jack on the monitor is an audio out – and subject to the model of monitor you have this may be the audio out connect for it.
In the case of connecting for wireless connectivity – connect the wireless transmitter to the rear of the monitor using the audio cable. Ensure that the device has live batteries and that the transmitter and the chair are both on the same channel. Where a wireless kit is being used this applies to the wireless receiver.