You certainly can! If you have that interesting message on the screen of your Kindle indicating it is in its charging mode, you will need to eject your Kindle (figuratively) from your computer, but keep the USB cable connected so it can continue to charge.
To eject your Kindle (in name only) if you are using a Windows computer, click the Start button, then click to open Computer. If your Kindle is connected via the USB cable, you should see it listed as an external drive and named Kindle. Right click on the Kindle icon and choose eject. You will see a little popup appear indicating you can now (physically) safely disconnect your Kindle. Leave the USB cable connected so it can continue to charge. If you look at your Kindle screen now, it should be back to normal and continue to charge while you use it.
To eject your Kindle (in name only) if you are using a Mac computer, drag the Kindle icon to the trash. You will see the icon change as you reach the trash and drop it in. Once it is gone, You will see the screen on your Kindle face return to normal. At this time, even if you keep the USB connected, you should be able to use it normally while it continues to charge.
These simple eject instructions should still work on the newest Kindles when they are in our homes and hands.