Where in MO? I've driven from Dallas to Chicago numerous times; the former is home now, the latter used to be.
If you're near KC, the easiest way is to probably just ride I-35 all the way to Dallas, and then hop I-45 from there.
If you're off of I-44 or near St. Louis, I'd suggest taking I-44 into Oklahoma, then taking US-69/US-75 all the way down to Dallas. It's primarily 4-lane highways, though it goes through a bunch of small towns. It's actually quite quick of a drive, with the added convenience of plenty of opportunities to stop.
45 between Dallas and Houston is relatively construction free right now; you might run into some around Houston, but nothing terrible. Traffic in both cities around rush hour can be epic, so plan your timings accordingly! There's really not much in the 3-4 hours between Dallas and Houston, so be warned...bright side, the speed limit is 75 a good part of the way.