For debian based distros (Ubuntu, Mint...) Download the following package: cncredalert_1.0_all.deb
For other linux distros please install wine:i386
After installing, start RA1installer.exe
RA1installer will take care of your wine/ddraw settings, there is no need to change anything.
You can also try PlayOnLinux
Download multi player (LAN/Skirmish/Online): CnCNet_Wineskin_Mac_OSX.zip
Download single player campaign: Porting kit
Alternative ways to install the game (complicated, only use PlayOnMac if the above downloads are not working)
Click "Install a non-listed program" at the bottom
Choose "install a program in a new virtual drive"
Click browse, choose RA1installer.exe and install the game
Enjoy the game!