There's really 3 possibilities here...
1) Both flights were booked on the same ticket. This is plenty of time, presuming your inbound flight is on time. You will NOT need to re-clear security - Alaska airlines uses Terminal 6 at LAX, and Air New Zealand uses Terminal B (AKA the Tom Bradley International Terminal). These two terminals are connected airside, with about a 10-15 minute walk between them. On a single ticket, your bags will be checked through to the second flight, so there is no need to collect them.
With both flights on the same ticket, if you do happen to miss your flight due to the inbound flight being late, then the airline will rebook you onto a later flight - although be aware that flight might be the next day (and they will generally not cover a hotel).
2) The flights were booked on Separate tickets, with no checked bags. In this case you will also have plenty of time, presuming you are able to check-in online. All international flights out of the US require you to be checked in at least one hour before flight. If you are able to check-in online the situation is the same as option 1 in terms of what you need to do.
If you can not check-in online, then you will need make it to the International terminal at least one hour before your flight, and hope that the Air NZ gate staff are willing to check you in at the gate (which they should be able to, but there's no guarantee!)
In either case, if you do not make it - including if your inbound flight is delayed - then the airlines will have no responsibility to rebook you on a later flight, and you will potentially be up for a significant fee to do so at the time.
Also remember that even if you don't plan to check bags, if the overhead bins on the flight are full when you board then you might be forced to check your carry-on bag.
3) The flights were booked on Separate tickets, with checked bags. Depending on the Alaska check-in agent at JFK, you MIGHT be able to have them check your bags through to the Air NZ flight - however on separate tickets they do not have to do this, you're basically at the whim of the check-in agent and whether or not they have had their morning coffee yet...
If they do check the bags all the way through, then it's the same as option 2 - you can stay airside, and go straight to the Air NZ gate, hopefully with either online check-in or enough time to check-in at the gate.
If Alaska refuse to check your bags through, then quite simply, you do not have enough time. You will need to go land-side, collect your checked bags, walk them over to Terminal B (10-15 mins), and then check-in for your Air NZ flight - all before the 1 hour limit passes. Whilst this is not impossible, it's very unlikely that you would be able to do it. Even then, you'd still need to get through security (which is generally an utter mess at that time) and get to the gate - all of which would likely be doable (presuming you made the 1 hour cut-off), but it wouldn't be pleasant...