The non-instrument rated private pilot obtained weather information from an online flight planning website. Prior to departure he also telephoned his destination airport and "determined the weather was all OK." He then took off and climbed to 5,500 feet above mean sea level for the cruise portion of his flight. He advised that he also checked and monitored weather while enroute, but the "fog and clouds came down really thick," so he decided to divert to the nearest airport. While flying towards his diversion airport, the visibility deteriorated and a "TERRAIN ALERT" warning illuminated on his GPS display. He attempted to pull up twice, but impacted trees and terrain on the side of a mountain. The pilot and passenger were seriously injured during the impact and the airplane was substantially damaged. The pilot reported that there were no preimpact mechanical failures or malfunctions of the airframe or engine that would have precluded normal operation.Review of weather observations indicated an extensive area of low clouds, and a large area of marginal visual flight rules (MVFR) conditions existed over the pilot's planned route. Low instrument flight rules (IFR) conditions prevailed at the diversion airport and in the vicinity of the accident site, with visibility less than 1/4-mile in heavy rain and overcast ceilings at 100 ft. The MVFR to IFR weather conditions had been forecast, and AIRMETs warning of IFR and mountain obscuration conditions had been issued.