Well the light link from SeaTac to King Street Station here in Seattle is really easy, though not terribly fast. It is a fun ride, just with a lot of stops. We did it once as we were coming back from a weekend away with my in laws and stayed downtown for the night without kids. It was a fun way to get to our downtown hotel and not deal with leaving our car at the airport. Not sure how the timing will work for you with flight times and Amtrak times but if you need to kill time by the King Street Station, there is a lot to do tourist wise and dining wise within walking distance. The light link gets off in the International district and then it's about a block walk to the Amtrak station I think.
On the way up, they check passports at the Vancouver Station. On the way back, they walk through the train, checking, as you cross the border.
In Vancouver, there is good information online about the routes on the light link. We rode it several times with no issues (downtown to Granville, checked out the olympic Village, ect) but it was a little limited in terms of where it went at that point and I've no idea how it may have expanded, or not, since then.
I did find this
"Pacific Central Station (the Amtrak Station) is half a block from the Main Street/Science World SkyTrain station where you can catch Vancouver's light rail service to downtown Vancouver. Trips take about 3 minutes and cost about $3.50 for adults and $2.25 for children. You can also take a taxi downtown for about $10.00."
The Expo and Millennium lines are the ones that go to Canada Place and the Waterfront Station which I think would be the one you want. Both of them stop at the Main Street Station so theoretically it should be pretty easy, other than thte two short walks you'll have, one in Seattle and one in Vancouver, to change train stations.
Now that I recall, we'd have tried this on our trip but it wasn't visible and it was pouring so we took a cab. LOL!
http://tripplanning.translink.bc.ca/hiwire?.a=iScheduleLookupSearch&LineName=999&LineAbbr=999
CL Klondike Kruisers
It is however very much like me. 
We are staying at the Cost Coal Harbour Hotel alone with several other cruisers. We are going ot spend Sunday and Monday exploring Vancouver. We got a great deal on the Hotel. In 5 months we will be on our way!
We will follow her anywhere.(except sushi).
