Is it the processor or grpahics card that handles x265 video?

Solution
Video decode is usually handled by the GPU. The Nvidia GPUs from the 600s and newer, and Intel integrated GPUs on Haswell (4xxx) and newer should be able to do hardware decode of h.265. Just make sure you've updated to the latest drivers. On older hardware, it'll end up being decoded in software as the hardware isn't set up to decode it.

Encrypted video streams like Netflix or Hulu are different. Hollywood insists that on a general purpose computing device like a PC, that those video streams must be decoded inside an encrypted virtual machine. Since the VM is encrypted, it can't send the video stream to the hardware on the video card. Consequently, decoding those videos is done entirely in software on the CPU.
Video decode is usually handled by the GPU. The Nvidia GPUs from the 600s and newer, and Intel integrated GPUs on Haswell (4xxx) and newer should be able to do hardware decode of h.265. Just make sure you've updated to the latest drivers. On older hardware, it'll end up being decoded in software as the hardware isn't set up to decode it.

Encrypted video streams like Netflix or Hulu are different. Hollywood insists that on a general purpose computing device like a PC, that those video streams must be decoded inside an encrypted virtual machine. Since the VM is encrypted, it can't send the video stream to the hardware on the video card. Consequently, decoding those videos is done entirely in software on the CPU.
 
Solution
It is a little bit of a mute point because you are using laptop which means both the CPU and GPU are not upgradeable to anything newer.

What you have can handle h265 just fine, what it wont be able to handle is 4k@60hz as it will be using HDMI 1.4 or earlier and thus only get 4k@30hz.
If this is only 1080p video then you will get full 60hz.