How to obtain no objection certificate (NOC) for visiting Neelum valley as a foreign national?

You don't make mention of how you're arranging your trip, or whether you've already applied for a visa. Should you be travelling to Pakistan either on business or as part of a tour group, you would be eligible for a visa on arrival. Otherwise, you would be applying for a visa at the Embassy in your home country, or country of residence.

The Azad Jammu & Kashmir (AJ&K) official government website for the tourism advises that, before visiting the area, foreigners should obtain the No Objection Certificate (NOC) through the Interior Ministry of the Government of Pakistan.

Similarly, tour companies which offer visits to the areas within the Azad Kashmir Line of Control (LoC) direct foreigners to make their own arrangements for the NOC, and to do so before entering Azad.

You make your application for the NOC (No Objection Certificate) at the Government of Pakistan Ministry of the Interior; note the last page lists the supporting documents needed.

Contact Information
Government of Pakistan Ministry of Interior
Room # 409, 4 th Floor R-Block
Pak Secretariat
Islamabad, Capital Territory
+ Tel: +92 (51)-9207544
+ Fax: +92 (51)-9202624

As having a visa is required in order to request the NOC to visit to Neelum Valley, you would want to mention that detail to the IO as you apply at the Embassy of Pakistan in Manama, Bahrain:

Chancery Address:
Building No. 35
Road No. 1901,
Block No. 319
Hoora,
P. O. Box 563,
Manama Bahrain
Telephone No: (+973) 17244113
Fax: (+973) 1725960, (+973) 17263807
Email: parepbahrain@mofa.gov.pk
parepbah@batelco.com.bh

Some Consulates may forward the NOC application to the Ministry in Islamabad, on your behalf, but that is not mentioned in the Embassy's Manama web information (as it is on its consulate site in the United States). Ask for guidance on whether it offers this service.