You would need minimum DisplayPort 1.2 or HDMI 2.0 capable gpu if you would like to run 2560x1440 165Hz on that monitor. Be sure to get powerful enough gpu power to play the games you want because the bigger the resolution the more gpu power you need.
And to use Freesync you should get a gpu that supports Freesync for example anything 1050ti and upwards from Nvidia and rx570 and upwards from AMD should work if they have enough performance for your games. For nvidia you might need to use DisplayPort cable as HDMI support with Freesync displays might not be available. For Nvidia GPUs you can download Nvidia drivers with similar functionality to Adrenaline from nvidia.com.
I don't know what gpu are you using right now to but if you are using the integrated gpu you have on that cpu I don't see any reason why it couldn't do 2560x1440 165Hz with Freesync if you just install the proper drivers and use proper cables and your motherboard connector supports DisplayPort 1.2 or HDMI 2.0. The gaming performance might be a bit underwhelming at that resolution if using the integrated gpu though.
You could use 2560x1440 resolution for desktop and enable Integer Scaling in AMD Adrenaline which will enable you to choose 1280x720 resolution in game. So you can get 4x the performance of 2560x1440 but you do lose on the picture quality a little bit.
Apply Integer Scaling Effects with AMD Software: Adrenalin Edition
