2. Half-Life isn't installed on this PC
You need to make sure of 2 things. 1, that you do have Half-Life properly bound to your account and 2, that you have actually downloaded the files for Half-Life, and played through the Training Course AS A BARE MINIMUM.
If you don't own Half-Life (the original version, NOT Half-Life: Source, or Half-Life 2), then you need to go out, and buy yourself a copy of it. It's carried by all good stores that stock video games, and is also carried by all major online retailers. You can also purchase it direct through Steam, using either a credit card (all of the world), or using a PayPal account (except in Germany, at the current moment).
If you do own Half-life, and have bound the CD-Key to your Steam account, then Steam will automatically download the files required to play the game.