With the iOne connected, it takes the place of your on-board sound card for both ASIO and Windows Audio (WDM / DIrect Sound) playback.
If the speakers have their own build in amplifier, you could technically plug the 1/4" output of the iOne into the Aux input on the computer speakers. You'd need to get a special Y cable from the music store (computer store doesn't sell this, it's called an Insert Cable) with 2 1/4" connectors on one end that go into the iOne and a 3.5mm stereo connector on the other end that goes into the Aux input on your computer speakers.
Most inexpensive computer speakers aren't going to sound very good for music production. These speakers are often EQ'ed for game playback, not music creating. Consider buying a pair of headphones to record and mix with that you plug into the iOne and /or get a pair of amplified speakers such as the Eris series speakers (for example) that are amplified.