First things first: what do you mean by “soon”? Virgin and Air NZ are ending their partnership on October 27. Unless something changes between now and then, you will have no chance of accessing Air NZ lounges by virtue of your Velocity status after October 27.
If you are travelling before October 27, my reading of the International Lounge Access page on the Velocity website is that you will not be able to access the Air NZ Lounge in Auckland while you are waiting to fly to LAX as it is not a Trans-Tasman flight. However, because Auckland-Melbourne is a Trans-Tasman flight, you will be able to access the lounge on your way home to Melbourne. Have a read of it yourself, but the eligibility rules seem pretty clear to me: “Lounge access is only available [to holders of Velocity status] when travelling on domestic or trans-Tasman services operated by Air New Zealand”. It says nothing about “on arrival” access which almost certainly indicates that “on arrival” Lounge access is not available.
On the other hand, there’s no harm in turning up to the Lounge before your LAX flight and asking nicely, particularly as you’ll be in transit.