When things are connected in series the same current runs through all of them. So it's 20mA not 20mA * 5.
And yes you would need a voltage at least great enough to allow 2V per LED. Voltages in series DO add up so that's 10V minimum. And because it's a really bad idea not to use a series resistor then you'd probably want at least 12V so you can drop 2V across the resistor to set the current.
Best practise is probably to run them in parallel with a resistor to each LED but a series connection with a single resistor can work for small low-current LEDs.
Steve