First thing is first, you need to connect your PS2 to a TV using the original composite cable (the yellow video cable) and go into the settings and change the video output to Y Cb/Pb Cr/Pr mode for Component output.
Then turn off the system and change out the cables for Component cables>
Component cables are the cables that have Red Blue Green for video instead of just one Yellow cable.
Next, you can use a component to HDMI adapter to adapt it to HDMI.
I use this adapter for my PS2 and my Wii on my HDTV as well as for use with my HDMI capture card
As mentioned above with the article, you can get a hold of component cables for the PS2 by shopping around.
These one's don't look too bad.
Amazon.com: TNP PS3 Component AV Cable (6 Feet) Premium High Resolution HDTV Component RCA Audio Video Cable for Sony PlayStation 3 PS3 and PlayStation 2 PS2 Gaming Console [Playstation 3]: Video Games
www.amazon.com
Adapting to HDMI makes it easier to connect the PS2 to all kinds of displays and of course HDMI capture cards.
And also if every other device you have uses HDMI, you can then plug it into an HDMI hub that goes into a single display.
It's up to you though, you can also go the other route of adapting to VGA and such as shown in the article.
In total, the HDMI adapter I linked is $40 and the component cables are $12 for a total of $52.
It's not going to get any cheaper than that going other routes with other adapters that adapt to other things.
Getting one of those chinese HDMI adapters that are these little flash drive sized boxes that plug directly into the back of the PS2 are not 100% reliable and whether or not they work, as well as whether or not they work for a long time before failing is always up and down so take your chances with those, and they aren't too much cheaper than going the route with cables and an Component to HDMI adapter.
I hope all this information has helped you to know what you need to be looking into so that you can adapt your PS2 to plug into your monitor.
Cheers.
P.S. - BTW, the cables may say they are for the PS3 but they will work on the PS2 just fine.