There are several problems that could lead to this. First and foremost, we need to check if your sound is on. To do this, while in the video press the topmost button on the side of the phone. This should trigger a display on your phone to show that the volume is going up.
If the display shows, but there is still no sound, make sure there is nothing in the headphone jack that could be redirecting the sound. This means that you can't have any headphones or charms inside the jack, otherwise the sound will be directed through them.
After trying both of those things with no luck, try to see if your speaker works at all. With the ringer volume all the way up, have somebody call you and check if the ringtone makes a noise. If not, the speaker is most likely the issue and should be replaced.
For further help try looking at the Troubleshooting page here: Galaxy Alpha Troubleshooting