It's best to go with what everyone else is doing and use your character's display name as opposed to their real-life name. If people see "Alex" written repeatedly in a post, it's counter-intuitive to everyone to suddenly start writing "Tim".

In the SAO light novels, the author avoids this confusion by always using "Kirito" (for example) to refer to the character while inside the game, and "Kazuto" while he's in the real world. Makes sense, since Kirito is the one doing the face-slicing in the virtual world while Kazuto is lying in a bed doing nothing in the real world.