The Command Console is enabled by default. If for whatever reason it is not, you first need to do some tweaking in your settings.ini of your save. This can be found in the "donotstarve" folder, located in the following paths :
Windows, Mac: KleiDoNotstarvesettings.ini
Linux: ~/.klei/DoNotStarve/settings.ini
Locate the settings.ini and open it with Notepad and where it says [MISC]ENABLECONSOLE = false and rewrite it as [MISC]ENABLECONSOLE = true then save the document.
Then to open the console while in game by pressing "~" by default on English keyboards. This can be changed at any time in the controls menu. Also you can hide the console menu again by pressing Ctrl + L.
You will need to run this command once to enable console commands:
RunScript("consolecommands")
Note: You have to re-enter every command after loading a world.