Each time you add a new computer with music in Itunes, using the "family ID" that music will be added to the cloud - if it is eligible. From then on, all "family ID" devices, your computers and your iOS devices (iphones, iPads) will see everything.
However, if you have music right now on an iPhone or an iPad that is NOT already in one of the computers you are going to use, then that music HAS to be put onto one of the computers. You cannot add music from an iOS device to the cloud, only download it from the cloud and play it.
KeithJenner is correct. Start with the computer where most of the music is and let it complete the matching process. Once it is done, go into your view options and check the "icloud status" box. Sort the view so you can see what matched, what was uploaded, and what was deemed ineligible. If you have ineligible music, you will probably need to make an AAC copy of them within itunes and run match again. Audiobooks, (videos other than iTunes purchased music videos) are not eligible at all.
Once you are certain you've got everything from the first computer matched or uploaded, then go to the second computer, sign in with the "family ID", and continue the process.
For more information, here is a page with all of the Apple support documents related to iTunes Match.
https://discussions.apple.com/thread/3619849