I currently am working on a website and have been trying to make a tiled background using this image of a tree.

I have been able to successfully tile the single image to create a full, high resolution background, but now I'd like to give it some movement. Is there any script or tool I could use to multiply the image of the tree and then make a gif of it moving down and to the right? Ideally I can make it look sort of like the gif below, but with the trees being bigger:

I'm very new to making gifs from images and I'm not really sure what the best method to approach this idea is. Any help or direction would be much appreciated. Thank you.
To reproduce the "yellow smilies" animated GIF effect with your tree icon, my approach would be to use the wonderful ImageMagick as follows:
There are command line definitions and examples available, and you'll no doubt have an entertaining couple of hours working it out. Don't try to solve all three steps in one attempt. Get #1 working. Then identify how to to extract an image region, and how to loop across the necessary range. Finally you can put the series together as an animation.