Greenwich to Hampton Court

Hasn’t Viking given you this information for boarding their ship? Google says train, ferry, bus, car.

This shows you where the ship berths- at somewhere called Greenwich Ship Tier- https://www.cruisemapper.com/ports/london-greenwich-port-11075#:~:text=Greenwich%20Ship%20Tier%20is%20a%20floating%20pontoon%20dock,and%20various%20other%20water%20taxis%20and%20riverboats%20dock.

"Greenwich Ship Tier is a floating pontoon dock in River Thames. Tender boats are used to ferry passengers between the boat and the pier. The Pier is on-shore and is where Thames Clippers (London City's river bus service) and various other water taxis and riverboats dock. "
This map shows where you step ashore-
https://www.google.com/maps/place/Uber+Boat+by+Thames+Clippers+-+Greenwich+Pier/@51.4839375,-0.0096414,17z/data=!3m1!4b1!4m6!3m5!1s0x48760347cbdffb71:0x5907fa0e30b11b84!8m2!3d51.4839375!4d-0.0096414!16s%2Fg%2F11h_t6gvv1

So to get to Hampton Court you need to get a main line train from Greenwich Station to London Bridge Station, change there for the Jubilee line to Waterloo, train to Hampton Court.
Total journey time= 1 hour 15 minutes.
Pay by Oyster Card or Contactless it will cost you £6.40 each way between 0930 and 1600, £9.40 at other times.
But you won't pay more than the £14.90 fare cap for Zones 1 to 6 in a day.

From the pier walk about five minutes to Cutty Sark DLR (Docklands Light Railway), take it to Bank

At Bank transfer to the Waterloo and City line to Waterloo

At Waterloo take a train to Hampton Court

The last part is not super frequent so whilst you could do the trip in 1h25 I would factor in 1h40 to cover imperfect train connections.

The route I gave is that suggested by DLR, and is the fastest route.
I would have taken a slightly different route from London Bridge to Waterloo, but that's what the TfL journey planner gives.
2 or 3 sensible routes.

There is a cheaper route in the fare system via Canada Water, and avoiding Zone 1, but that is a round the way route, DLR to Canary Wharf, Jubilee line to Canada Water, Overground to Clapham Junction then train to Hampton Court, or DLR to Shadwell, Overground to Clapham Junction, train- both of which are long way round routes, not recommended for a visitor.
So there are at least 5 routes. TfL shows at least 4 others which avoid Zone 1, of increasing and bewildering complexity, and hints at 2 others.

I'm glad to see The Drain getting some publicity... I love it.,.

The Waterloo and City doesn't get any love around here - It is worth it just to claim another Line and be able to brag, "I was on the The Drain today".

There is even a boat route back along the Thames- 3 1/2 hours from Hampton Court to Westminster, then the Thames Clipper straight back to Greenwich in 40 minutes.

As you can see, there are numerous ways to go from Greenwich to Hampton Court. (Recently, on numerous threads, we've seen a "my way is better than your way" spat even though all answers are perfectly fine.) The right way is the way that is right for you.

My suggestion....go one way to Hampton Court and return via another. If you have the time, and the weather is nice, take the boat one way.

Also be aware that there are no ticket barriers at Hampton Court. If you are using an Oyster card or contactless you will need to tap out. The machines are on the wall near the entrance to the tracks.

If you are intending to take the boat back from Hampton Court (which is a very interesting and worthwhile ride) it is best to pre-book it- it does sell out, as there is usually only the one afternoon boat which comes all the way back to Westminster.
Some return boats only go as far as Richmond, from where you can return by train or tube to the boat, making the return journey shorter but still giving variety.
If you do that, just look at a tube and train map, and work out a way back to Greenwich, as there are too many possible routes to list, and it is of no benefit to do so. Especially given Frank II's usual and totally un-necessary barbed comment.
Part of the fun of a really comprehensive and dense transport system like in London is just finding a route that suits you and finding /seeing the unexpected.
As you are docked for two days you have the luxury of not rushing back to the ship.
If you look up National Rail, as I did last night for the fares, they give you a different set of fares, and a different route again (not listed above, but would be my first choice route), and that suggests buying the slightly more expensive Zone 1 to 6 off peak day travelcard at £15.20.
For 30 pence either way it doesn't make any real difference.
You would do that if you wanted to have the flexibility of using a bus at some stage in the day.

https://www.thamesriverboats.co.uk/

Since you are coming from the ship, is Viking offering any excursions to Hampton Court? That might be the easiest way.

FYI--the Cutty Sark DLR station is about half the distance to the ship than the Greenwich train station.