Hey dub, I fish the lower stretch of the tualatin quite a bit, and I have also kayaked the lower couple of miles (as well as the middle/upper stretch). I've never been to River Run city park, but it looks like it's on what I call the crappy part of the Tualatin (not that it's bad fishing, just much dirtier and slower). River Run is probably a few miles above the dam which separates the slow part from the fast part of the Tualatin. If I were you, I wouldn't go too far down from that park because the dam, from what I've seen on google maps, doesn't look passable. For above the dam, refer to bubs post, but you're only a 5-10 minute drive from the lower T, so I'll let you know all I can about that...
The main parks on the Lower T are Fields Bridge, Swiftshore, and Willamette (in order from higher up all the way down to the mouth). The visibility is much higher and the water moves a lot quicker on this part of the Tualatin. It's mostly just rapids and big pools (with a lot of 2-3 foot deep riffles), and coming from someone with experience, a kayak doesn't really help you get that much access. Most of the good holes are accessible by bank. Not to mention, unless you're used to kayaking in faster water, this one is tough, with a couple 5-6 foot drops and some rapids. As far as fish go, I have only seen bass and coho in this stretch of the river. I'm guessing it's too warm for trout, but it looks like it could hold them and I might try sometime. In comparison with the upper T, there are fewer species of fish, but there's a lot of them. I've had luck fishing senkos for bass and I've seen others fish nightcrawlers effectively too. The coho fishing should be just picking up in the lower T, but it does get somewhat crowded at some of the parks, and taking a kayak down to the mouth at Willamette park is your best bet if you're coho fishing (or just going up to one of the many parks on the mid/upper T, I know bubs fishes around Hillsboro).
Anyway, good luck!