There are several possible approaches, but each one has limitations, and all of them provide far less flight ability than using internal combustion Otto, Diesel or even Brayton engines (piston engines and jet engines, respectively)
If steam power is used as flight assist (i.e. for a glider like aircraft to lift off), then two techniques come to mind, either storing steam in a high pressure container and using like a JATO rocket to launch the aircraft (much like Bob Truax did with Evil Knievel's Snake River Canyon "motorcycle"), or have the high pressure steam used to run an engine just long enough for takeoff and climb to gliding flight. The second option leaves the boiler and firebox on the ground, but does not subject the aircraft to violent acceleration.
Direct production of steam without a boiler through chemical reactions is possible. The WWII German Me-163 rocket plane was powered by high pressure steam created by mixing highly concentrated Hydrogen Peroxide with a mixture of methanol and water. The fuel and reaction itself was insanely dangerous, but if you're looking for a rocket powered interceptor, this might be the way to go. A somewhat less dangerous reaction was using "Walter" technology, where Hydrogen Peroxide was used as an oxidizer for more conventional engines. Walter turbines were fitted to seem experimental Germa n U boars towards the end of the war, buring a mixture of Diesel fuel and concentrated H2O2. The combustion products and steam expanded in the turbine, giving the submarine rather spectacular performance, however the short range and danger of using concentrated H2O2 as the oxidizer was a negative factor for the Navy.
A final option would be to ditch the steam altogether and use the Stirling engine. This is also an external combustion engine, but uses a gas contained within rather than water as the working fluid. While early Stirling Engines were quite heavy, they were designed as stationary engines. There have been many lightweight designs that were developed by car companies in the 1960s and 70's, which could be adaptable for aircraft usage. Once again, the power to weight ration is lower than a comparable internal combustion engine, but a Stirling engine is quiet and economical, and in a large "steampunk" aircraft there should be room for multiple engines, the large radiators and external heaters needed.
Junkers G-38. Imagine filing the wing space with Stirling Engines