It's do-able but I'm not sure that it's really advisable. If you do it, the only way it would work would be via this route. When you're not on the these roads are narrow and windy so don't expect to make very good time. It would be a gorgeous drive though. However, you're not going to have much time to explore if you try to do this all in one day. If you leave at the crack of dawn, you should be able to get in a couple VERY short hikes, but nothing much. You're also missing Paradise, probably the more popular part of Mt. Rainier. Sunrise is also spectacular though. Anyway, there's a lot of great stuff all along this route so you're going to be rushed and have to skip some amazing places. Any way to make this a two-day trip?
Whatever you do, don't skip it!! Way too beautiful (Rainier) and amazing (St. Helens) to miss, imho.
If you have to skip anything, I suggest it be Snoqualmie Falls. But, really, once you do the two mountains, the zig over to the falls really isn't adding all that much time or miles. Why don't you see how late it is and how tired you are aftering visiting Mt. Rainier before you decide whether you are up for seeing the falls? If you decide you're not up for it, your route would look like this.
How many days are you staying in Seattle? Snoqualmie Falls is really just a very short daytrip from there.