Use the console commands recordnetdemo demo_name, recorddemo demo_name,  playnetdemo  demo_name, playdemo  demo_name and
  playdemo stop. Example:

Multiplayer (Smaller Demo Files)
recordnetdemo uc_vs_uc // starts demo recording
stopnetdemo // stops demo recording

playnetdemo uc_vs_uc // starts demo playing
stopnetdemo stop // stops demo playing 

Singleplayer
recorddemo UCLevel1Tips // starts demo recording
stoprecording // stops demo recording


You can also save hard disk space while recording demos by setting com_compressDemos to 0, 1, 2, or 3.

0 - No compression, creates large demo files
1 - Fast compression and decompression, creates small/medium demo files
2 - Slow compression and fast decompression, creates small demo files
3 - Fast compression and slow decompression, creates medium size demo files

You can also use the command compressdemo to compress a demo 

See the FAQ How do I access the console? for details on how to enable the console.