NOTE: This how-to may enable you to download paid/cracked apps, for which I take no responsibility or liability - do at your own risk.
First, you need to download APKtor, which is available on the Market. You can also download it from http://apktor.gotoinit.com/download/ either using the browser on your phone or from a computer, then transferring the APK file (APKs are Android app installation files) to your device and running it using a file manager (I recommend ES File Explorer grab it here if your Market is not working).
After installation, start up APKtor. You will be notified that no repositories are setup yet. Repositories (aka repo's) are URLs pointing to app collections. Once you add some repos you can directly download and install apps from APKtor. The best way to find repos is to Google around for them.
After adding some repo's APKtor should automatically populate a huge list of apps. You can force an update by pressing the Menu/Options button (left button under screen) then touching Update. Simply touch an app and then touch install. You can uninstall apps the usual way and also by going to the "Installed" page within APKtor, touching an app and touching uninstall.