Android - How do I access my phone (not my SD card) from my computer?

You could install the Android SDK on your computer. This would give you the possibility to access your device's storage via the command line using e.g. adb push and adb pull. If that's not the level of comfort you want, there are also graphical tools utilizing ADB:

ADB FileExplorer is a minimalistic frontend to copy files from/to the device. It's freely available from the XDA developers (simply follow the link). It is written in Java, and thus should run on nearly all operating systems used on desktop computers.

If you want a little more comfort, then there's also ADBBrowser. Available for Linux and Windows (32-bit and 64-bit), it offers also an app-browser. And last but not least, there's QtADB available for Linux, Mac and Windows (again 32-bit and 64-bit), full of useful features like file manager, app manager, device information, SMS, shell, screenshots, logcat, backup/restore... Take a pick.

Of course, there are many more possibilities. If your device is rooted, there are several apps available on the Playstore which permit full access to the local file system remotely.