Should a حجّة be written for my father, who passed away before performing the Hajj pilgrimage after winning the lottery and obtaining his passport, or must I perform the Hajj pilgrimage on his behalf?

If the deceased had not performed the Hajj of Islam and left behind wealth, then someone must be hired to perform Hajj on their behalf from their estate, before settling debts and distributing the inheritance. If they did not leave behind wealth, then Hajj is not obligatory on their behalf, but it is commendable, and the one performing Hajj on their behalf must have already performed Hajj for themselves.