This article has some of most common steps and resolutions to why the client will not open when you run the program. Make sure that you complete every single step, as every single one is vital in hopes of resolving your issue.
If you are still unable to see the wonderful patcher screen with the almighty giant orange button that says "PLAY", then please do not hesitate to let us know here in Player Support by submitting a ticket. We can work with you to determine why this is happening, and how we can resolve it quickly. If you do send in a ticket, include these logs: