Do you have the passive Alesis monitors? They made active ones too.
How you connect the amp to the computer depends what audio connections your computer has. What is the computer / sound-card?
I'm guessing your computer has a 3.5mm stereo jack for audio output, like a headphone plug. The Alesis RA300 has RCA (aka phono) inputs so you need to shop for a 3.5mm stereo jack to rca cable I guess.
http://www.ebay.co.uk/itm/3-5mm-Ste...t=LH_DefaultDomain_3&var=&hash=item540a287849
You will also need a way to control volume. The RA300 amp I think has two volume controls on the front, one for each speaker which is a bit of a pain to use. You can also adjust volume output from the software on the computer. I'd recommend keeping the amp turned down away from full volume and use the software control around 70% most of the time. Otherwise, if the software accidentally goes to full level you will blow your ears off!