Depends on how you choose your numbers. Let's say you choose five different numbers. For the first die, the chance of getting a number on your list is 5 in 6. For the second, the chance is now 4 in six. Continuing, you can see that the probability of getting all five is 5!/6^5 = 120/7776.
But what if you duplicate numbers? If you pick all ones, there is only one solution in 7776 that meets that requirement. I think the general formula will looks like:
5!/7776/n1!/n2!/n3!/n4!/n5!/n6! where nx is the number of times you chose x as a number.
So if you choose 2,3,3,4,4, n1=0, n2=1, n3=2, n4=2 and n5 and n6=0 so you get 5!/7776/1/1/2/2/1/1 = 30/7776