As has been mentioned, turn at a speed you are comfortable with handling, but make absolutely CERTAIN that the bowl blank is properly secured. Put the tailstock live center right up to the bottom of your bowl to give you additional support while rounding out your blank... turning between centers is always more secure than turning an unsupported blank.
You generally won't be using a dovetailed tenon chuck (typically a 4-jawed chuck) before rough turning the outside of the bowl blank to ensure that it is turning on center and not out of balance ... Use the live center on the tailstock for as long as possible ... generally till it's actually in your way.
Once that is done, then you sand and basically finish the outside and do most of the finishing work on the base of the bowl and turn your dovetail tenon. To assist in gripping the bowl by the tenon, it needs to be at least 3/8ths of an inch deep, I think, and the walls around it (to the outside of the bowl) should be at least an inch wide for proper strength to support the bowl while you are gripping it from inside the tenon hole. It may seem like a lot, but if you have the extra wide face plate for your chuck with the rubber gripping pins, you can do additional work to remove an oversized tenon at the end. Of course, if your bowl is pretty small, the strength requirement of the tenon will also be lower.
Then, you take the screw chuck or faceplate off and turn your bowl around and attach it with the 4-jaw chuck, making sure that it is securely fastened.
DO NOT STICK PAPER TOWEL OR OTHER MATERIAL IN THE DOVETAIL TO PROTECT YOUR BOWL FINISH. This would seriously reduce the grip your chuck has on your work piece, greatly increasing the risk that it will fail to turn in a stable fashion and probably cause you injury when it flies loose. I would use a forstner drill bit in a tailstock chuck to remove the center of the blank down to around your target depth and to hog out the majority of the wood to be removed, if possible, and then finish turning, sanding, and finishing the interior and the lip of the bowl.
After all that is done, you have the option of turning the bowl around one last time to grip it by the rim with wide flat jaws on a 4-jaw chuck so that you can fine tune and finish the very bottom of the bowl, or you could just finish it by hand as many others do, if your tenon is small enough to not be of concern.
In all cases, use of calipers to ensure you are not making your bowl's walls too thin is a very good idea, use of faceplate to protect your face is pretty much a requirement, and use of at least eye protection should be mandatory. Respiratory protection is also a very good idea to practice, especially while sanding.
There are some very good videos out on google about turning bowls. I'll recommend watching Captain Eddie makin some shavings!
