Global challenge mode is a portion of Grid 2 online where you compete with your friends and rivals in a series of challenges that randomize every week. Each challenge has three scores or times that that you can beat to earn a gold, silver, or bronze award. To unlock this trophy you must be in first place on the leaderboard that displays when you enter global challenge mode, you don't need to have a gold score or even complete every event that is posted (and if you dont have DLC you won't be able to anyway as there is always one global challenge that requires DLC), as long as you are first on the leaderboard you will be fine. But completing each event with a gold will increase your chances of being first on the leaderboard.
There is a chance that it might not pop up when it is supposed to at the end of the week so make a backup save before checking the global challenge section once the week is over. If it doesn't popup, delete your save and replace it with your backup file. Keep in mind that it might take a few tries but it will eventually popup.