There are different visualization techniques that can be used to obtain high speed images of the combustion process.
The easiest to implement (less expensive) will be shadowgraphy (or the more sensitive version, Schlieren). This techniques uses the fact that density variations inside of the chamber (due to the flame) alters the index of refraction of the medium causing the light rays to bend (it is commonly known in nature as the mirage effect). Shadowgraphy allows you to capture images of the density gradients inside the combustor. For this technique you'll need a bright LED, a few lenses and mirrors, as well as a high speed camera. I recommend reading GS Settles's book for reference: " Schlieren and Shadowgraph Techniques".
Another one which is also relatively simple is chemiluminescence. The camera captures light emitted by electronically excited radicals inside of the chamber. For example, you can do OH* chemiluminescence and this will allow you to visualize the flame front (since OH* is always located in the hot region very close to the flame front) and you can extract info such as flame speed, stretching and flow regime (laminar or turbulent). The only thing you need here is a bandpass filter that you put in front of the camera. Since light is emitted naturally by the species you don't need a light source.
There are also laser techniques. These are more sophisticated and require more specialized equipment (not only a laser source but also, in many cases, an intensified high speed camera).The ones worth mentioning are: Rayleigh scattering, planar laser induced fluorescence (PLIF) and particle image velocimetry (PIV). Camera intensifiers are very expensive in case you don't already have one and the lasers are high power and can get very specialized also.