AxeFX III is itself an interface. For basic home recording you don't really need a dedicated interface unless you want more features like not needing the axefx powered on to use your headphones/monitors, a software mixer, more mic preamps, etc.
Personally I like having a dedicated interface since my Fractal unit is not always connected and I like having more control over my routing and monitoring via a dedicated software mixer, for example I have a dedicated fader in my interface's control software for my DAW's metronome, faders for each input on my interface, etc.
In your situation I would start by getting a hardware mic preamp and connecting its balanced output to one of the axefx's balanced inputs. Doesn't have to be anything fancy, something for a few hundred bucks like a Useful Arts Hornet will give you a lot of flexibility to dial in "color" i.e. harmonic distortion that you won't get from clinical interface mic preamps if you ever get a dedicated interface down the road.
Your other option is to get an interface. Even something like the Scarlett 8i6 will give you a software mixer, lower impedance headphone amps, better monitoring flexibility and routing, and a lot of I/O, including SPDIF so you can record and reamp the axefx in stereo with identical quality to USB. The Scarlett mic pres aren't going to measure up to high end stuff but the air mode is nice if you want a little character on the mic (high boost and some harmonic distortion).