The problem you have is that VGA cables trasmit analogue signals. HDMI is digital. What you'd need is something that converts analogue to digital, which requires active electronics and is a pretty complicated process. They exist, but they'll be more expensive and quality will be entirely dependent on how good the analogue to digital converter is.
Newegg have a few, this one's less that $30:
http://www.newegg.com/Product/Product.aspx?Item=N82E16812186218&cm_re=VGA_to_HDMI_converter-_-12-186-218-_-Product
I have no idea whether it's any good.
You can't go VGA-> DVI -> HDMI. VGA to DVI works because DVI-I includes support for both digital and analogue signals. The DVI -> HDMI adapters only send the digital signal (HDMI is digital only), so those adapters would drop the signal being sent from the VGA port.
How old is the computer? If it's a desktop, it might be worth grabbing a bottom-of-the-range video card, something like this:
http://www.newegg.com/Product/Product.aspx?Item=N82E16814130815
That's just $30 after MIR, will output a proper HDMI signal (with sound, if necessary). It won't do any gaming, but it's a perfectly capable for desktop, youtube, netflix, etc.