1) If your Mic is USB mic, you don't need a external mixer. Plug your Mic to computer USB port directly.
2) If your Mic is not USB mic.
a. If Mic need Phantom power (e.g. Condenser Mic), you need to connect it to a Mixer/Audio Interface (with Phantom Power Supply) first. Use a Fire-wire or USB cable to link your Mixer/AI to your computer.
(Note: Assume your Mixer has USB or Fire-wire interface)
b. If your Mic doesn't need Phantom power (e.g. Dynamic Mic), you can choose either way. But make sure your Mic is able to plug in Audio card of your computer. Most of cases, you need a 6.33mm -> 3.5mm Mono TRS converter.
c.If you have Audio Interface or your Mixer has USB interface, I would recommend that you plug your Mic to Mixer first, then use a USB cable link your Mixer to your computer. By doing this, you will have lower noise and less latency(Delay) recording.
Good luck and enjoy!