You cannot access the properties of an object while you're in the middle of defining it with an object literal. If you want to base the value of one property on another, then you must either do it after or use an intermediate value.
For example, to do so after:
<<set $player = {
"name" : "Lupen",
"hp" : 50,
"ap" : 15,
"str" : 5,
"dmg" : 0,
"dex" : 6,
"int" : 10,
"cha" : 10,
"desc" : "Itsa me, Lupen."
}>>
<<set $player.dmg = $player.str * 2>>
Alternatively. Here's an example of one way to use an use an intermediate value (using a temporary variable):
<<set _str = 5>>
<<set $player = {
"name" : "Lupen",
"hp" : 50,
"ap" : 15,
"str" : _str,
"dmg" : _str * 2,
"dex" : 6,
"int" : 10,
"cha" : 10,
"desc" : "Itsa me, Lupen."
}>>