I do not believe this is possible:
TL; DR: The submarine will be far, far, too slow and cannot easily be weaponised by a Roman.
A diving suit with Soda Lime CO2 extractor might work, but I have not covered that.
Water pressure
The Romans certainly could build ships. It makes sense that a Roman shipwright could build a vaguely submarine-ish, underwater vessel. Since the ship will be underwater, however, it will have to withstand a slightly higher water pressure. Wolfram Alpha tells me that the pressure 10 feet below sea level is 130kPa. As Mike L
points out in the comments, a wooden hull would bend and flex underwater, just like a regular ships hull. As the hull bends seams will briefly open and the submarine will leak. On an ordinary ship, this water is pumped or bucketed overboard, but you can't really do this on a submarine. The sub would sink and not be able to surface. You could make the submarine out of a "watertight all-metal rigid hull" I doubt the Romans could easily cast a hollow submarine sized piece of metal. Making the ship out of smaller metal plates would introduce the same problems as a wooden ship.
Manoeuvring
Another problem would be manoeuvring. Modern submarines dive by taking water into special compartments and surface by removing the water from those compartments (usually by pumping in a gas). This would be difficult for the Romans to replicate because they did not have the sort of precision engineering required to make such a device, nor do they have access to pressurised gas to flush the dive compartments.
Overcoming Friction
Disclaimer: this section is entirely based on my 10 minute google. If you (the reader) know what your talking about then I have no problem with you improving this with an edit. Hint hint.
One thing the other answers do not cover, is that moving underwater is hard. This physics.SE question and this linked site helpfully give the following equation for finding the drag force exerted on your submarine:
$$F=\frac{1}{2} ρv^2A c_d$$
We will assume that the front of the submarine is spherical (assume a spherical submarine in a perfect vacuum...) with a radius of 2.5 meters. If I understand the table from the second link correctly, then the drag coefficient ($c_d$) of the hull should be about $1.1$ (or between a human and bundle of wires - which is my (very bad) estimation for a wooden sub).
We have:
$c_d = 1.1$
$ρ = 1000$ (density of fluid, $1000kg/m^3$ for water)
$v = 3.3$ (Target velocity, $m/s$, ~ 2 times walking speed)
$A = 2\pi2.5^2 = ~39.27m^2$ (Area of the front of submarine)
Which gives us:
$$ F=\frac{1}{2} 1000 \times 1.1^2 \times 39.27 \times 1.1 $$
Or about $104536$ Newtons of resistance to travel at walking speed, which is insane. A single human can push around 630N with a firm surface to push off, on land.
Even if my calculation is outrageously wrong (it must be), I do not believe a single Roman or small group of people would be able to propel the submarine at any speed.
Roman ships had many oars. The trireme for example had 170 oarsmen. If we put all 170 in the submarine described above, they would have to exert 615 N of force each to maintain a speed of $2.2 m/s$ ie: slightly more than walking speed.
Air
With 170 men aboard, your ship will need lots of air. I do not believe that the Romans could have compressed air, at least not to the pressures required.
Soda Lime might be enough of a few people: (Quote from my early draft)
As the other answers have pointed out, removing CO2 from the ship is your primary concern and you can do this with Soda Lime, which is made from Calcium hydroxide (limewater, which the Romans had access to) and Sodium hydroxide (aka caustic soda, aka lye). Making Soda Lime is well within the Roman's grasp.
But I realised, that's just not going to cut it with 170 men. You will either need a funnel and bellows for air (better be a big funnel) or a lot of Soda Lime.
Weaponisation
The trireme relied on speed to ram enemy ships becaue the Romans did not have cannons or other convenient point and shoot weapons. Some ships had trebuchets. Ramming is out of the question for the sub, as are underwater trebuchets, so weaponisation will be difficult if not impossible for your Roman general. Dronz
points out in the comments that an underwater ballista could be used, point blank. This is possible but the performance of the ballista will be affected by water, so the final speed and power of each bolt would be limited. Weaponising the sub will still be difficult.