Q:

Why did Killmonger bother with the museum?

So at the start of the movie, Killmonger and this character are working together:

 Ulysses Klaue.

Eventually though

 Killmonger kills Ulysses and uses his body as leverage to gain the Border Tribe's favor or something.

But now I'm wondering:

 Why did Killmonger go through all the trouble and danger of robbing the museum? Why didn't he just kill Ulysses at first glance and immediately head to Wakanda?

A:

TLDR

This is hard to answer without spoilers...
Assuming he knew something (from his father or other sources) about the time frame it would take for T'Challa to be crowned after his father's death it is very likely that:

 He was trying to draw out King T'Challa because usurping the throne would look better if the current King failed before he succeeded.

So as we learn during the movie, Killmonger is

 A kick ass military might that uses deadly and merciless tactics to achieve his goals. According to Everett K. Ross at some point in the movie, he was part of a team for the CIA that started coups and used pinpoint assassinations to upend countries to make it easier to manipulate them into doing things that they want.

Given this:

 Killmonger knows usurping the throne is easier if you can prove that you are able to accomplish a task that the current King was too weak to complete. This is why Killmonger set up and ensured that T'Challa failed his mission. Had this not been his plan why would he have stopped him? Therefore, drawing out Klaue triggered the Wakandans into action, additionally all those affected by his actions (W'Kabi) were seriously looking for him to be brought to justice. Everyone also knew that this was the best (and possible last) chance to get him.

And:

 After T'Challa failed, and the anger and depression of the failure had a chance to set in, Killmonger went straight to W'Kabi (I am sure that he had current sources to tell him current dealings of the Wakandans to know that this was the village to go to and W'Kabi was the one to talk to...) and laid the body at his feet. This was the best and easiest way he had to be delivered straight to the throne room to issue his challenge as well as ensure he would have at least some loyal support after becoming King.

A:

Like all good evil genius plots, this one is multi-staged:

 1) Set up the bank robbery to draw Klaue out of hiding.
 2) Allow Klaue to get caught.
 3) Help Klaue escape so Wakanda feels let down by leadership.
 4) Kill Klaue.
 5) Bring Klaue to Wakanda and boast, "See what I did that your all powerful leader could not?"

If Killmonger just skipped directly to step 5, he wouldn't have the "I'm better/I'm the true protector of Wakanda" air about him.

