PC David Rathband was found dead by police at his home in Blyth, Northumberland, on Wednesday night.
His brother Darren flew back from Australia and on Saturday, he was at the weekly Newcastle Parkrun, which his brother took part in.
He said he would continue the work of the Blue Lamp Foundation, which his brother set up.
Runners in the 5km run at Newcastle's Exhibition Park held a minute's silence to remember PC Rathband
The police officer, originally from Stafford, set up his charity after he was shot twice by Moat in Newcastle in July 2010.
It aims to help emergency service staff injured in the line of duty and the target is to raise Â£1million by 2013.
In a post on Twitter, Mr Rathband said: "Makes me weep reading all your messages, broken but determined to keep David's foundation growing for others. Please help me."
In other posts, he asks people to support the Blue Lamp Foundation.
PC Rathband was hailed for his charity work and took part in a number of fundraising events including last year's London Marathon.
Following news of his death, many people paid tribute to him including Prime Minister David Cameron, former Home Secretary David Blunkett and Northumbria Police Chief Constable Sue Sim.
On Friday, the inquest into PC Rathband's death heard he was found hanged.
The twin brother of the police officer shot and blinded by Raoul Moat said he would carry on the work of his charity.