Back up all data to at least two different storage devices, if you haven't already done so. One backup is not enough to be safe.The backups can be made with Time Machine or with a mirroring tool such as "Carbon Copy Cloner." Preferably both.
Boot into Recovery mode, launch Disk Utility, and erase the startup volume with the default options. This operation will destroy all data on the volume, so you had be better be sure of your backups. Quit Disk Utility and install OS X. When you reboot, you'll be prompted to go through the initial setup process. That’s when you transfer the data from one of your backups.
Transfer only "Users" and "Settings" – not "Applications" or "Other files." Don't transfer the Guest account, if it was enabled on the old system. Test. If the problem is still there, the cause is external to the machine, or else you're dealing with a bug or a hardware fault.
If the problem is resolved, reinstall your third-party software cautiously. Self-contained applications that install into the Applications folder by drag-and-drop or download from the App Store are safe. Anything that comes packaged as an installer or that prompts for an administrator password is suspect, and you must test thoroughly after reinstalling each such item to make sure you haven't restored the problem.