It's not specified in the question, so I'll assume that it's a gasoline-powered fixed-wing SEL airplane (aircraft is anything which flies, including balloons, rotarywing, etc...) on a standard climbout from a standard (but looooong) runway, etc... I'm also assuming you can't do exotic things like change the engine compression ratio.
I will also assume that efficiency in this context is only energy consumed per unit height. If we want to define efficiency as energy consumed per unit distance traveled, then the problem moves in a different direction and becomes much more complicated.
In my responses below, please understand that these are first-order effects, but second-order effects can lead to modifications. For instance, maximal efficiency might come at a slightly faster/slower airspeed in order to allow for a slightly faster/slower engine RPM.
Also, in general, fuel is not the important consideration for takeoff. Safety is. It's better to burn a few mL more of fuel and keep the airplane from disaster (which would presumably result in wasting a lot more energy).
Mixture (with important caveat at the end)
The mixture is the biggest control on efficiency. In order to achieve maximal efficiency, you need to run as lean as possible. Preferably lean-of-peak, if this can be done. We're talking 20% savings or more here.
If you're not taking off at high altitudes, e.g. in the mountains, don't ever try this if you value your airplane engine's life, and as a consequence your own. The engine needs the enhanced cooling from the rich fuel/air mixture. Only at high density altitudes can/should the engine be run at full power at extremely lean settings.
The throttle should be fully open, which will provide the least restrictions to the engine breathing.
If you have a constant-speed prop, then it should ultimately be run as slowly as possible. This gives maximal time for the burning fuel energy to be extracted, as well as minimizing the number of engine cycles and friction.
The RPM should be advanced at such a speed as to provide peak propeller efficiency for the advance ratio.
Use the lowest octane gas you can, as this has more energy content than higher octane. Please note that lower octane increases the propensity of the engine to knock/detonate.
Fly the plane at Vy, which is best climb airspeed. This is the airframe's most efficient operating point.
In general, don't use them for optimal efficiency. They increase drag more than the increase lift. That's why we like them on landing because we can come in at a steeper angle of attack.
An argument could be made for rapidly "pumping" the flaps in order to pop the plane off the ground, thus eliminating rolling resistance at an earlier point in the takeoff roll. This technique was employed to great effect at a recent STOL competition.