My experience with LXC is that it's cool when it's working, but it turns to a nightmare once something stops working, especially in the network. Want to fix it? Prepare your altar for the sacrifice of a virgin and never forget to continously throw up gang signs! (Something it has apparently inherited from it's de-facto predecessor, OpenVZ.) Using LXC would also limit you to use the kernel of the host system, since it's containerization, not virtualization.

Gaming in a virtual machine requires the ability to pass-through PCI. Xen and KVM both support that, with a catch - you need a second graphics card, because the one dedicated to the virtual machine is unusable for the host system. I only dabbled with it shortly, worked pretty well while using a GT640 for the host and a GTX980 for the virtual machine. I found this in my bookmarks, maybe it helps.

I would go with KVM. It's deeply integrated into most Linux distributions, has awesome management tools such as virt-manager, all while being easily scriptable.