By default BCM will allow you to drag your chat frames to the very edge of the screen, the remaining customization is done in BCM's modules which can be enabled or disabled at will.
Type /bcm to configure BasicChatMods, a basic and modular approach to chat customization.
Currently provided by Blizzard, right-click combat log tab > Settings > Formatting > Show Timestamp. Repeat this for every combat log, e.g. "My Actions", "What happened to me?".