Based on the fabric and one I have that my mother did in the very early 1930's...it is 1930-ish. However, the thing about quilts is that they were often made from old fabric...Grandmother's apron, cousin's shirt....etc. I know my mother made this quilt out of new fabric...but your quilt may have been made later. It appears to be machine pieced? That so?
How you wash it depends on lots of things....what the stuffing is? how thick it is and how big the quilt is and how precious it is to you, what all the fabrics are and how likely the colors of that quilt underneath are to run...to bleed color into the top quilt....how great your anxiety is about it falling apart.
The most gentle way you can clean it is to first make a nylon netting bag for the quilt, then soak it in a solution or Orvis soap ( available at quilt shops)...bounce it up and down to get water through it and soak at least 24 hours...a bath tub is best for this. Then drain the water and refill....always cold water, of course...bounce more to rinse and repeat....pushing down on the quilt to get the soap out. NEVER lift a wet quilt by an end. The weight and the wet weakened fiber is very likely to tear. Finish your rinsing with distilled water (rain water or water from a dehumidifier is good) and rinse until litmus paper tells you the water is no longer acedic.
Then drain the tub, press as much water out as you can, lift using the netting and cut open the netting and lay on many towels, direct several fans on the quilt and allow it to dry. Use the netting to lift the quilt and replace the towels with dry ones.
Never hang to dry....the weight of the wet quilt will likely cause tears.
Good luck....that's a lot of wet stuff to deal with!!
Linda C

Oh boy, I never thought about the quilt underneath bleeding through. The patchwork underneath looks much older, maybe even Victorian based on comparing the fabric prints to a book I have about textile history. It's very dark colors, all navy, black, and maroons- I don't think it's worth the risk of it bleeding. Maybe I'll just let the kitten have it for a bed/playhouse instead of trying to use it on a bed. He has pretty much taken it over, anyway! Thanks for your help, Linda! If I ever find myself with another antique quilt, I'll know exactly how to clean it.

The quilt underneath may well be much more valuable then the one covering it up. It may have been "out dated" and simply used as filling for a newer machine stitched quilt used to keep a sleeper warm.
Can you 'un sew" the top quilt and see what's beneath?

Here is a link that might be useful: Crazy quilts

Good advice.

That's what I was thinking, Linda - if there's a Victorian quilt under there, then I'm displaying it and the cat can just find another fort to play in! I think it may be possible to unsew it, as the top layer is held in place only along the edges and every so often with a knotted stitch that can probably be undone easily. If I'm able to get it apart later this week, I will post photos right away for identification.

Anxiously awaiting.....then you will have 2 quilts!!