Ideally, as per rules, you should apply for DPC right when you submit the online application and dispatch the hard copy to FPSC. On the top right corner of page 1 of your application, you should label 'Advance Application' (meaning app through proper channel will follow). Ideally speaking, you should submit your DPC *with* the hard copy of your application, but FPSC knows that usually it is a slow process and not possible to get DPC in such a short period of time. So they allow you to submit your DPC uptil April~May, i.e., before the announcement of result of the written exam. If you are claiming age relaxation, they make sure that you submit your DPC well before result announcement, else, they reject your candidature. My DPC was accepted in November~December and one of my friends even submitted in May (he wasn't claiming age relaxation, just completing formality).


TLDR
Best to get it ASAP and submit to FPSC while retaining copy with you. If that isnt possible, get it and submit it to them latest by may or before written result announcement (whichever is earlier).