Check which kind of 1/4-inch jack you need. The cables and jacks are available in both mono and stereo. A stereo plug will feature two black bands around it, while a mono plug will have only one band. There is also a third type, the high-end TRS (Tip, Ring, Sleeve) jack, which provides a balanced connection for audio or balanced microphones. Jacks of this third type are often unique to the equipment they support, so it is best to check the manual for wiring reference.
Remove the outer shielding from the audio cable if necessary, and identify the necessary wires. You will either have a mono, two-wire cable, featuring signal and ground wires, or a three-wire stereo cable, with ground wire and a left/right pair. Lay the cable down next to the connector plug and check the lengths of the exposed wires. There should be enough to reach all the contacts, but not so much that any exposed wire will be visible once the connector cover is screwed on. If your cable has a screw-on metal shield, be sure to slide this onto the cable before you start soldering, or you won't be able to fit it at all!
Strip around 5mm of insulation from the tips of each wire, and loop them through their appropriate connectors. In a mono cable, attach the signal wire to the tab that protrudes from inside the end of the cylindrical plug; the signal should be transmitted from the very tip of the connector. The ground wire should be soldered to the common pin that touches the exterior of the connector. The wiring is the same for a stereo pair, only there will be a left and right connector, rather than a single signal pin. It is possible to use stereo connectors with mono cable; the right signal connector is simply ignored.
Allow the solder on all wires to cool before checking each connection manually. Gently move the wire back and forth to ensure that there is a solid connection with adequate contact. If any solder has run down the connecter, clean your soldering iron and remove it. Any bridging of connections will prevent your cable from working, and could be dangerous.