To get the PU you have a few options :
- Mission, if you form GB you will get the mission to PU France for exemple (dlc ?), these mission CB last 20years so be prepared before clicking it.
- Waiting, if the king dies without an heir there's a chance that you either inherit or get yourself in a PU war, see Atwix guide for great info on this part
- Claiming the throne, if a nation with your dynasty and a royal marriage have no heir or one with a low claim, you can click the claim throne and get the CB to PU them. You have to act quickly as you can lose the CB if the situation change.
The one with the more player agency is the third option but it also present more risk of losing a new PU compared to the second option. Compared to vassals, with PU you want to focus more on improve relation bonus instead of diplo reputation as liberty desire have little consequences, but you absolutly need to get your relation above 0.
If you are planning to get the PU's by war, always check your ruler's age before going for one. Seems quite obvious but easily forgetable. It would be a shame to battle France for 5 years and lose the PU a month later because your king is old. Have a good amount of ducats available to be able to gift and influence the new PU. Improve relation, have the estate with same religion +25 if available, give subsidies, have the advisor or diplo idea that imrpove relation. If you are the curia controler or the HRE leader use the appoint cardinal or bestow imperial grace for extra relation.
Do you have more precise questions ?