How do I prevent smoked brisket from being chewy?

Chewy means undercooked. Most of your standard "barbecue cuts" of meat contain a lot of connective tissue. This must be rendered to achieve tenderness. This goes for brisket, pork butt, and ribs, to name a few. If you are using the words "chewy" or "tough" to describe the texture of your meat, in nearly all cases it has not been cooked enough. Your time/temperature pretty much confirms it.

The best thing you can do is deemphasize time and temperature as your measure of "done." Rather, use a skewer to probe your meat when you think you are getting close. It should slide in and out of the brisket with VERY little resistance. Some people like to wrap in foil a few hours in. This will finish the brisket faster, as you will mitigate the evaporative cooling effect that causes your cooking process to stall. Plus, you can add some liquids and spices to the foil, which adds to the flavor profile of your meat. This is better known as the "Texas Crutch," and there is no shame in using it. Plenty of competitions have been won employing these tactics.

Brisket is a little intimidating because of the small window between "too chewy" and "dry and falling apart." But you should probably err more on the side of the latter, as that can be more easily compensated. The real takeaway here is have patience, and look to the meat itself for clues as to whether or not it is done, rather than using a timer and thermometer.

Here is a good basic technique for smoking brisket. The important points are to smoke it until it reaches an internal temperature of about 160 F, then wrap in foil. The meat won't absorb any more smoke flavor at that point anyway, and the foil will protect it from drying out during the rest of the cooking process. You can add a little bit of liquid inside the foil, though I've never bothered.

After foiling the brisket, continue cooking until it reaches 195-200 or so. Remove it from heat and allow it to rest for a while. Some people will store it in an insulated cooler.

Properly cooking brisket can take 12 hours or more. If you can't manage that kind of time, think about smoking chuck roasts ("chuckies") or making pulled pork.

As the other answers point out your brisket was vastly undercooked. Understand that this style of cooking requires not only cooking to doneness but also "time at temperature" in order to break down the connective tissue.

Some poeple do inject marinades, but that's not where you are going to find the gains you are looking for here. Your biggest opportunity to improve this dish is to cook much longer. Generally at least 12 hours at 225 degrees F. If you want, you can begin on the smoke and then move to an oven to finish. The general rule of thumb is that you don't get much more smoke adhesion after hour 2 or so.

The reason I wanted to add on to the existing answers was because I do not see a mention of slicing technique. Be sure to cut your finished product against the grain for maximum tenderness. This is one of those things that makes a huge difference but people are often skeptical about unless they've seen it first hand. Check out Franklin's slicing video on youtube for one surefire method. This whole series is great. It was produced for PBS (American public television).

I sear both sides of the brisket, wrap it in foil, and allow to cook until done, occasionally spreading liquid runoff over the meat.