Aharon, as the first Kohen, was the quintessential representative of what it means to be a Kohen.
We see (Bamidbar 25:11-12) that Pinchas was awarded with becoming a Kohen for being קנא את קנאתי - taking up My (Hashem's) zealotry. So Kehuna fits with taking up Hashem's zealotry. As the prime alpha Kohen, Aharon certainly had this attribute.
Aharon chased Shalom even though he loved Shalom. Had he despised Shalom, then he wouldn't have been chasing Shalom for Hashem's zealotry - it would have been his own. It is only by loving Shalom that you can then chase him and be fulfilling the lofty level of taking up Hashem's zealotry.
Alternatively, it refers to chasing Shalom in the race for more reputation points. You should love him for all his contributions to the site, and chase after him in pursuit of more repuation. Gershon Gold is the only known violator of this precept.