You don't check-in your bags to CGK. The agent will understand that you do a turn around at CGK. And your bags should only go from SIN to MNL.

By doing a return check-in, you will get your BP for SIN-CGK as well as CGK-SIN. This will make you be able to do a quick transit without going through immigration and customs. Once you get off the plane follow the sign for the immigration. Just before the immigration, you will see an escalator up, and prior to that, if you look to the right and left, you will see transit desk.
Before the escalator, there is an x-ray. Sometimes there is a person who man the x-ray. From many times I have done this in the past, generally my carry -on need to get x-ray. I still have time to go to the lounge (Esplanade Lounge) to wait for the flight. Once it's announced boarding, I head to the gate. And remember that you need to pay the departure tax. You can pay in Singapore Dollars or USD, but of course the exchange rate would not be favourable.