If your headphones and mic each use 3.5mm connections, you might first try getting two sets of splitter cables like these, one for the headphones, and the other for the microphone, and seeing if that does the trick...
https/www.monoprice.com/product?p_id=669
I noticed one of the reviews there actually mention someone using this for hooking up a single set of computer speakers to both their Xbox 360 and a PC, and that may work for an Xbox One as well, although someone else mentioned having the volume get reduced for their PC when powering off their PS3 hooked up in a similar manner, probably due to the signal getting grounded. It likely depends on how exactly the equipment is designed, though if it works in your scenario, it would definitely be the simplest and least expensive option. You would need a second splitter for the microphone, and would probably want a pair of 3.5mm male to female extension cables like these as well, unless the ports on the two devices happen to be right next to each other...
https/www.monoprice.com/product?p_id=648
Also, if getting a splitter from somewhere else, make sure it's the same kind, since many of these tend to have the plugs in the opposite direction, with two plugs and one jack, for hooking two sets of speakers/headphones to one device. There are also some that look just like this, but they're for splitting a combined microphone/headphone jack into separate microphone and headphone jacks. Make sure it's one that simply links a stereo signal two ways.
And yeah, if this happened to not work, an amplifier that lets you switch between two input sources might, though I'm not sure what you would need to switch over the microphone input.
(I was writing this post earlier, but had to leave before I could finish it, so there was another reply in the mean time with a similar solution. : ) )