Your player's bard wants to perform a concert for the king and his court to earn their favor. Do you ask her to roll a skill check for Performance or for her lute as a tool? I'm thinking Performance... but then when she use her lute proficiency?
Your player wants to influence the king, so the check is charisma (Performance.) It's not just the king, though - the entire court is present, and rather than make a check for each of them I would simply set the DC a step higher. First, however, I would ask whether the bard was trying to use his mastery of the lute to impress the king and court and thereby aid his performance check.
The bard will be able to play well without a check. If he wants to help his performance, though, I'll need a basic description. Successfully playing something hard (DC 20) might get him advantage on the eventual charisma (Performance) check. Successfully displaying virtuosity by playing something very hard or nearly impossible (DC 25 or 30) would also lower the DC of the performance check, and might carry an additional benefit based on the circumstance.
If this is a major story beat in the campaign, however, you might consider running it as a 4e style skill challenge. Set a DC based on how difficult, in general terms, you think it will be to influence the royal court (see the Conversation Reaction table, DMG pg. 245) and ask the party members what skills they want to use to earn the favor of the throne. Obviously the bard can use dex (lute) and cha (perform,) but the cleric can also use wis (religion) to put the party's requests in a context favorable to the kingdom's divine patron, the wizard can use history or investigation to point out the ways in which favoring the party is good for the kingdom generally or the king and courtiers specifically, and one or more insight checks can provide advantage to the other skill checks. You can determine the end result based on the number of successful skill checks the party gets before 3 failures: less than 3 gets them kicked out; 3 - 5 get them begrudging acceptance, or perhaps support from the king with opposition from several courtiers; 6 - 8 gets them what they asked for with some measure of enthusiasm; 9+ gets them a royal charter and free access to the castle armory.
My personal belief is that the more significant a thing is to the story, the less it should just come down to a single simple die roll. If it matters to the campaign whether the party earns the royal favor, then you want to encourage using as many skills as possible. If it comes down to just the bard, then both the lute proficiency and the performance skill should be on the table, as well as insight to read the crowd and history to pick the right song. Once the bard has earned advantage on the performance check and knocked a few points off the DC through other skills, then it becomes much more gratifying when he rolls two ones and experiences crushing defeat.