I booked our June cruise with HAL directly.
What are our chances of an upgrade (to a better wheelchair room)? Anything I should be doing?

when you made your booking you would have been asked if "you would like a complementary upgrade should one become available"... most likely youre marked as not wanting one BECAUSE if you're in a wheelchair accessible room and say that you want an upgrade you may be placed in a room that is not wheel chair accessible.

youre better off leaving the booking as is and sailing in the type of stateroom you paid for. or you could always call and see if there are any other staterooms available that are WCA and pay for an upgrade....