My computer is broken, how can I get my music onto my iPad?
If it was music you purchased from the iTunes store it is automatically available for you to redownload from the iTunes Store on your iPad. You don't need to have bought iTunes Match; tracks purchased via the iTunes Store are always available for re-downloading fromt he store. Just set your iPad up with your iTunes account and you should see the tracks in the store as having be bought and available for download to your iPad.

If the music wasn't from the iTunes Store and you weren't using iTunes Match to copy your music in to iCloud your options are extremely limited. With iTunes Match you could turn on the feature on your iPad and you would see your entire library, stored in the cloud, available to you.

If the hard drive of your broken computer is still functional you can hook it up to another computer, associate the new machine with your Apple ID, copy the music off the old computer's hard drive by using iTunes Add to Library feature found under the File menu. Once you've added the files to the library, you can connect your iPad to this machine and transfer the music on to your iPad. At this point you may want to consider subscribing to iTunes Match to move the content to the cloud, otherwise you need to use this computer to sync new music to the iPad.

If the hard drive of your broken computer is no longer functional you have no options. Your files are lost. Hopefully you had a back up in the form of Time Machine or a third party service like CrashPlan.