One way is to use a StringVar. When you change it's contents, the change shows up on the GUI. And you should read up on buttons as yours don't do anything when clicked.
from Tkinter import *
from tkFileDialog import *
class Game():
def __init__(self,root):
self.root = root
self.var1 = StringVar()
self.var1.set("D")
btn = Button (self.root, textvariable=self.var1, width=5, command=self.change_var)
btn.grid(row=1,column=1)
def change_var(self):
var=self.var1.get()
if "A"==var:
self.var1.set("D")
else:
self.var1.set("A")
root = Tk()
root.title("Sample Application")
root.geometry("100x50")
Game(root)
root.mainloop()