How to properly assemble a flyback transformer?
How do I connect the Primary First coil and the Primary second coil to
  obtain the correct electrical diagram?


Look at the bobbin below and the pins I've marked in red, green and purple: -



Option 1


Connect Primary First to red pins and 
Primary Second to purple pins
Join pins 2 and 4 on the PCB.


Option 2


Connect Primary First to pin 1 and pin 3 (green)
Connect Primary Second to (same) green pin and pin 5 (purple)


In these options be sure that the direction of winding on the bobbin remains consistent on all windings.

You can of course use pins 6 to 10 as an alternative to pins 1 to 5 for the primary windings.


For the secondary winding, use pins 6 and 7 or 9 and 10.
For the tertiary (auxiliary) windings use pins 9 and 10 or 6 and 7 i.e. the spare pins left after choosing the secondary pins. Leave the middle pin free to promote extra clearance.


As the primary wires come from the bobbin to the pins it might be good practice to sleeve them to prevent criss-crossing primary and secondary wires rubbing against each other.

You might also consider a 90% coverage in copper tape around the finished windings that can be grounded on (say) pin 8. If ground isn't used as an electrical connection you can usefully tie the screen to the negative supply of the bridge rectifier as a means of shunting away some noise.