It's going to depend on a lot of different factors. Fiber content will determine type of dye, and even then the stitches around the seams and such may likely react differently to the dye. If the jacket is a natural fiber like cotton, it is dyeable, but if the thread used for seaming is synthetic, it will not take they dye unless you use a dye designed for synthetics, and that is a fairly intensive process involving a pot and boiling and constant stirring for a considerable length of time (I could make a Cajun roux in less time, and that's saying a lot!)
You might try asking a tailor or dry cleaner in your area for suggestions, or see if you can find a "fiber artist" near you because they often know how different fibers will react to dyes and can check the fiber content and give you specific suggestions.