You've not mentioned if you are booking the trip through CP or AA (using a travel agent or not), but this could be important. Your best option is a CP booking with CP flight numbers on each segment. This means you just need your single black "utimate upgrade" for both segments. You just call CP Gold desk and tell them what you want to do. They'll take care of the rest and check availability from upgrade inventory. You may have better luck on the NYC-YVR than YVR-HKG, and if you do not get the HKG, confirm your upgrade using a grey sticker and save the black until the day before, or day of the flight, to see if space has opened up to HKG and they can confirm you all the way.
However, if the NYC-YVR is booked as AA, then the following applies:
Remember, there's no guarantee of an upgrade, though your itinerary will require a variety of stickers. Assuming from your colour references that these are last year's and valid through Feb 2001. To see what flights have likely availability, check out the ITN process. This has been explained many times, on this and other fora, so pleased go back 60-days and check out any ITN postings.
First, your AA flight between NYC-YVR. Again, at 72-hrs prior to your AA departure call the airline that issued the tickets (or through whose res system your travel agent booked the flights) and indicate you are Gold and want to use the "old" stickers. The AA flight will require 2 grey system-wides or 4 500-mile stickers. AA limits system-wides to max of 1,000-miles. (I assume NYC-YVR is about 2,000-miles distance.) You can then also request the YVR-HKG on CP by telling the agent you are using the system-wide black sticker.
Because you are flying two separate carriers, unless the AA flight is done as a CP codeshare, you won't be able to use the black "ultimate upgrade" for the entire set of flights out and back. And if the CP flight is booked as an AA codeshare, you won't be able to use it at all, since AA upgrades are for domestic or transborder flights only. This is critical, so it would likely be best to get your ticket via CP not AA.
Also, make sure you are not on an AC codeshare CP flight, operating with an AC aircraft. (These will have at least four digits, versus the CP flights which will have just one.) If so, you won't be able to use last year's upgrades, and can only fly with this years on the restricted list of full-fare economy tickets.
You also didn't mention if you were renewed as Gold through 2001. This is important, because if you reverted to Club -- or basic -- you can only pre-book upgrades 24-hours ahead (Club) or standby at gate (basic). While non-Gold and EPs can still use the domestic stickers (reds and greys), the "ultimate" may be restricted to the top two elite tiers, which means if you did not requalify for the current year, you may not be able to use them at all.
EMPRESS does this routing (YVR-HKG) quite a bit, so can probably offer you some tips. Sure he'll be heard from later today. Good luck.