Modern mortars carry their propellant in these little ring-shaped charges that slip into the tuby end of the mortar shell, just behind the fins. A percussion cap on the end sets off the charges, and the mortar man can adjust his range by removing and adding ring charges.
Building a black-powder mortar isn't too hard, most of the ones I seen built are machined from cold-rolled 1026 or 1018 carbon steel stock, which is pretty much low-carbon steel which have been cold-worked to increase its strength. Though you have to watch your barrel length, with the example you posted above, I think in 18th century standards that would had been considered a howitzer or even a short field cannon. Are you sure this isn't just a modern military mortar refurbished for black powder? The longer the barrel, the higher the firing pressure. That being said, one should use the coarsest grain black powder one can find, Fg being the minimum. But with a safe charge and a barrel length less than 15 calibers, an antique artillery piece shouldn't have to expect to take more than 10,000 psi of pressure when firing.
I haven't built one myself, but I done some reading into the subject. Some of the builds out there use seamless DOM tubing for the barrel, so no need to bore out from a solid block of metal. Still need a lathe to cut the threads for the breech plug though. C1026 DOM with would be the minimal steel grade, chrome-moly 4130 or 4140 tubing would be optimal. For a 1.5 inch bore barrel, a C1026 DOM tubing with 1/2 inch wall thickness ought to hold. You can calculate the pressure rating for a cylindrical tube by using Barlow's Formula:
Maximum Working Pressure = (2 x Wall Thickness x Yield Strength) / (Outer Diameter of Tube x [Safety Factor]) <--- For Safety Factor, I would use at least a 2
So for C1026 DOM with a Yield Strength of 60,000 psi, a 1.5 inch ID (2.5" OD) tube with a wall thickness of 0.5 inches would have a maximum working pressure of 24,000 psi ---> 16,000 psi when building with a safety factor of 2.
NOTE: Work pressure would be the maximum pressure the tube can handle before it starts to stretch permanently. The maximum pressure before bursting would be higher, and can be found by substituting yield strength with ultimate tensile strength, which would be around 80,000 psi for C1026 DOM. Of course, you shouldn't build according to this number, since repeated firing over the maximum working pressure would stretch the barrel thinner and thinner until it eventually bursts.