Its probably the converter then. Was it a cheap one? I bought a cheap display port to VGA one off eBay and it was pants, everything was blue. Spent a few quid more and got an HP one which works fine.
When the screen goes black to you get any errors in event viewer in either windows system or...
How good is the laptop and what's the resolution that you're trying to output to your TV?
It could be that the graphics chip is not quite up to the task.