Can I build a transistor through two diodes?
You just can't. There's a transistor-model which is explained with two diodes and you can measure the two diodes in the transistor but you cannot make a transistor with two diodes. 

The NPN transistor has two pn-junctions, same as your diodes, but in the transistor there's only N-P-N, in your diodes you have N-P-metal-breadboard-metal-P-N. When switching on the transistor, there are electrons going from emitter to the base. The base is very thin, so most of the electrons diffund directly to the collector. The assymetrical doping of collector and emitter and the very thin shared base differs the real transistor from your diodes.