What is the best way to clear the harder trials in SSFIV on PS3 without a fightstick?

A lot of hardcore fighting gamers will tell you that you have to have a stick to be any good. You can ignore those people. As part of any company's standard testing routine, you must be able to do anything in the game using any of the supported input methods (controller, fight stick, fight pad, etc.). Not only that, but there are tournament players who win using pads. The trials are definitely possible.

What I would recommend if you're having trouble, is looking up a full trial run on youtube for your character. Sometimes just seeing how the combo is supposed to look is what you need to complete it. On top of that, a good way to master an extended combo is to break it into small parts (like single links or cancels) and then slowly put the pieces together.

As motivation (I hope), follow a video showing Vesper doing some trials on a normal pad.

This is the video description.

This video is a tribute to all pad players. I started out as a pad player myself in 1993 on the Snes playing Street Fighter II Turbo. No matter how much people bash pad players all combos and tournaments are achievable through practice and dedication, not accessory.

Super Street Fighter 4 Trials - Controller Pad