Is your backyard rink sloped?
It is always preferable, of course, to have a yard that is as level as possible since it will be easier to work with, but the large majority of yards that I have seen have a certain slope. It is almost rare to find a perfectly level yard, without shallow "bowls" or crevices all over the place.
My backyard as well has a 6 to 7 inch downward slope at one of its corners, and as you can see from the photos on the site my ice surface is very level, so don't let it stop you from building your rink. I achieve smooth and level ice from the end of November to mid-March!
If your slope is bigger, or if your yard has shallow "bowls" as at different points, it is not a reason to prevent you from building your rink. They are definitely concerns that you will have to account and plan for, without the need for extra cash for a bulldozer to level your yard (also an option).
Any body of running water moves down-slope under the influence of 1) gravity and 2) the curvature of the Earth's surface, so here’s what I would do:
2) I would make / buy rink boards
that are higher on the downward sloped side than the boards that you
install on the higher sides of the yard -- assuming that you are using
boards that are 2 feet in height all around. If you are using 4 feet
high boards you might be able to make all the boards the same size, but
will be a little more expensive.
At flooding, the water will fill up the deeper end and the shallow "bowls" first. The higher boards will allow for the accumulation of water. The liner will prevent the water from draining out of your yard, and when the weather drops below freezing, your rink will begin to freeze.
You would need to add 1 or 2 inches of water during the night, let it freeze and add more water the following night, repeating the process until the water level has covered the entire yard, the entire ice surface is level, and the minimum ice depth at the thinnest point is around 3 inches.
No doubt, the thickness of the ice will vary at different areas but this should not become a worry. You only need maybe 2 inches of solid ice at any point. However you do need leveled ice and I think the above method would be the right way to do it.
I hope the above information helps!