As several others have said, using a standard drill bit to enlarge the chambers will get you in trouble.
The two flutes cutting edges and the rapid twist will bite into the steel and attempt to pull the drill bit rapidly into the hole.
If the cylinder isn't held tight enough it will start flailing around damaging itself and everything in sight.
If you do clamp the cylinder securely there is a good chance you will damage it by crushing it out of round.
Even if the drill bit doesn't grab the part it will probably produce a hole that is about as rough as the ones in the cylinder now.
The proper way of enlarging the holes is by either reaming or honing.
Reamers typically are designed with straight flutes that won't pull the tool into the hole.
Although the cutting edges on the front of each flute will bite into the steel they won't pull the tool into the hole.
Reamers are designed to enlarge a hole about .005 (five thousandths) of an inch of material, total.
Keep in mind that although each cutting edge of the reamer is only cutting off .0025 thick chips there are at least 4 and often 6 flutes at work so some real power is needed to drive it.
Reamers tend to be self piloting but they do need to be kept square with the hole while they are first starting to cut. After they do, they will follow the hole quite nicely.
Due to this tendency, reamers used in drill presses are rarely forced to a fixed position. The floating drive head allows them to follow the hole freely but transmits power to them.
Using a crossing T drive handle similar to a large tap wrench will also accomplish this when the hole is being hand reamed.
As I mentioned, reamers are designed to enlarge the hole only about .005, if more enlargement is needed either several different reamers are needed or an adjustable reamer is required.
Most adjustable reamers are quite a bit larger than the .310- .316 size we're talking about.
Reamers, properly lubricated and used will produce very good, accurate finishes.
Now, as I mentioned, honing is often used to enlarge holes and to improve the surface finish.
Honing is a grinding process so lots of lubrication is needed to wash away the removed material and the grinding material off of the tool.
It just so happens that automotive valves often have valve stems that are 5/16 inch in diameter.
If you've got some money and want to play around trying to improve the condition of the chambers you might go to your local Auto Supply and buy a Valve Stem Hone.
They don't require holding the cylinder too tightly and they can be used with a electric drill.
If you go this route, do not try to enlarge the chamber all the way to the bottom. Just concentrate on the first 1/2 inch of the holes depth.
The hone is designed for a thru hole so the bottom of the chamber may damage it.
Here's a link to show you what they are:
http://www.goodson.com/240_Grit_Standard_Duty_Flex-Hones_in_14_Stock_Diameters/