It was sometimes a practice for the new king to declare a general amnesty for all condemned criminals during a coronation. Lots of adventure hooks if, say, a murderer with a grudge is once more loose in the city, or the city guard needs help just scooping them all back up again.
On the other hand, it was also popular to have executions. Since the new king is evil, there could be all manner of hangings, beheadings, burnings, and impalings.
A lot of extra religious services get held on coronation week. The churches/temples would be full all day with masses and liturgical ceremonies of blessing. Maybe free healings will be offered from sunrise to sunset on the actual day.
Kings often give gifts to the city populace...to encourage popularity. Step right up and take your pick of a shiny silver piece or a nice, healthy piglet (the evil uncle intends on taxing it right back anyway).
Tournaments are absolutely, historically, and fantastically obligatory. If your setting isn't strictly medieval, a gladiatorial combat against vicious monsters would be a crowd-pleaser. The players could sign-up for one, or rescue slaves sentenced to fight in them, or kill some of the monsters that get loose, or find out who's rigging the fights.
David Copperfield and Chris Angel square off in a competition to see who is truly the most powerful wizard, and who will serve as court mage to the king.
Doomsayers fill the streets, prophesying the king's evil reign. They all disappear the next day.
Each night, a coven of well-dressed vampires attend one of the many balls that take place throughout the week. They don't seem interested in eating anyone, and the nobles are actually more fascinated by their charm than frightened of their reputation.
An ominous astrological event takes place the night before the coronation. Meteors fall like burning rain into the city, or a blood red comet lights the night, or a new constellation of piercing green stars appears in the heavens.