How can I jailbreak and unlock an iPhone 4S running iOS 6.1?

To jailbreak an iPhone 4S running iOS 6.1, the best tool to use is evasi0n. This is an untethered jailbreak that is compatible with your firmware version. You should first back up your device using iTunes or iCloud. Download the evasi0n tool on your computer, connect your iPhone, and follow the prompts to complete the process. Regarding the unlock, there is no software-based utility like ultrasn0w that works for the iPhone 4S on this firmware. You should contact your service provider to request an official factory unlock. If a carrier unlock is not possible, you may need to use a hardware SIM interposer like an R-SIM or GPP card, though a factory unlock is always the more stable and permanent solution.