So you have Twitter account from some time. Wether it is your personal or company account it is good to personalize the look and feel to represent your selfhood or image of business. Of course there is nothing wrong about having default background all of Twitter’s predefined images are lovely and clean.
1. Repeating pattern
This technique is probably as old as internet. We just need small image and it will be repeated to fill whole background. If we will craft one special tile we can achieve endless pattern. This kind of background will fill display with every resolution.
Create your own?
There are virtually no limitations regarding designing Twitter background. Only rule to remember is that column with tweets is fixed width and regarding viewers resolution more or less of background is visible.
You are more than welcome to be inspired by my own Twitter background.