To download the latest beta build of the game (PC only) click on the following link.
http://www.machine22.com/humanextinctionsimulator/beta/hesPreview1.zip
To start to play simply unzip the files and launch HES.exe. Don't forget to vote for Human Extinction Simulator on Greenlight!
You can do so by visiting the official forums here: http://www.machine22.com/forums/
Please provide as much details as possible and specs of your PC is reporting something unrelated to gameplay (screen resolution problems for example).
Human Extinction Simulator has a unique way to handle movements and attacks of units so it can take some time to get used to it. Don't worry as after playing a few missions you'll fully get the hang of it. To help you get started here's a video demonstrating how to complete the first simulation in 4 turns (spoilers ahead!).
Documentation for modding will be posted in the forums (currently a work in progress). Be warned that as long as the game is in beta that you might have to adjust your mods as new versions are released.