Actually, I am a grad student, so I am free all day everyday except for my supervisor's weekly group meetings.
I would like to write software for intraday trading algorithms, but I am not sure why I won't have to build an automated trading system.
What are you looking to do exactly? HFT or just trading in general but via automation. Both the trading algorithms whether intraday or daily or for longer periods and automated trading system are of the same. They are both trading automatically via machine codes but HFT is more of trading mechanism and automated trading system is more a structured set of trading rules.
But if you are looking to do HFT, forget it, $20K is NOT nearly enough. If it's trading in general, for some instruments, $20K is enough.