There will be a write up of the methodology in respectable terms in the final report, but it's essentially knowledge+google search+checking links on various sites (such as those found by the above processes, and some of the effective altruism sites that have looked into this).
It is not uncommon to find organisations working, directly or indirectly, on a single Global Catastrophic Risk (GCR). For instance, the World Health Organization does much work to prevent pandemics, as part of its remit.
It is rarer for organisations to focus on multiple GCRs - for a start, this involves them having the concept akin to GCR, which is not often the case. In a report I'm preparing with Dennis Pamlin of the Global Challenges Foundation, here is a list of organisations focusing on multiple GCRs (note that it is not necessarily an endorsement of their quality). Let me know if there are any organisations missing, and I'll add them:
Brookings
http://www.brookings.edu/research#topics/
Bulletin of the Atomic Scientists
http://thebulletin.org/
CSER
http://cser.org/emerging-risks-from-technology/
Center for International Security and Cooperation
http://cisac.stanford.edu/
Club of Rome
http://www.clubofrome.org/
Council on Foreign Relations
http://www.cfr.org/issue/
Federation of American Scientists
http://www.fas.org/programs
Future of Humanity Institute
http://www.fhi.ox.ac.uk/research/research-areas/
Global Catastrophic Risk Institute
http://gcrinstitute.org/research/
Institute for Defence Studies and Analyses
http://www.idsa.in/topics
International Risk Governance Council
http://www.irgc.org/issues/
Lifeboat Foundation
http://lifeboat.com/ex/programs
Nuclear Threat Initiative
http://www.nti.org/
Saving Humanity from Homo Sapiens
http://shfhs.org/whatarexrisks.html
Skoll Global Threats Fund
http://www.skollglobalthreats.org/
Stimson Center
http://www.stimson.org/programs/
Risk Response Network
http://forumblog.org/communities/risk-response-network/
World Economic Forum
http://www.weforum.org/reports/global-risks-2014-report
Tower Watson
http://www.towerswatson.com/en/Insights/IC-Types/Survey-Research-Results/2013/10/Extreme-risks-2013
What kind of GCRs are being worked on? This can be seen in the following graph; AI is not as badly situated as I would initially have expected: