Screen attachment for Nikon D3300?

Is it possible to purchase a side screen that faces towards you when pointing the camera at yourself that connects maybe via the HDMI or A/V ports on the side?

Yes, those screens are called "field monitors"

B&H Photo and Amazon sell them

Also search for "on camera monitors". You can also use a regular computer monitor with HDMI input. If you don't need the small size, a computer monitor is probably cheaper. Plus, a computer monitor is larger which makes it easier to look at.