As with a fair number of the Doctor's enemies, the basic initial reason for the enmity was that the Doctor opposed its efforts to dominate & to control others -- & (critically) that he did so effectively. It's the Doctor's effectiveness that marks him out as a special enemy.
In "The Name of the Doctor", we saw that the Great Intelligence went after all of the Doctor's previous incarnations, not just the Second. The difference between its efforts against the Second Doctor & those against the other incarnations is that, in 2 incidents ("The Abominable Snowmen" & "The Web of Fear"), the Second Doctor was able to identify the GI as being behind them. In most other cases, the GI's involvement was successfully hidden.