I am playing a barbarian (battlerager) with fighter levels. I will end up Champion 4/Battlerager 16.
I initially chose fighter to be able to afford spiked armour, since I was not sure the campaign would let me buy some once it started, and barbarians do not start with enough gold to purchase it. But, I also appreciated the Second Wind and GWF, the former very useful at very low levels (and much less useful later on).
What I most wanted about the Champion was the improved critical, which I felt meshed very well the barbarian's escalating brutal critical trait. Since barbarians get extra dice on crits, doubling the crit chance was a no-brainer, and taking GWF meshed with wielding a greataxe and taking the GWM feat.
Action surge also doesn't hurt at 2nd level fighter. It is not a full replacement for getting extra attacks, but it does compensate a little bit for it.
The tough decision was whether to take the 4th fighter level or not. In the end, I felt the ASI or feat was worth it. Otherwise, taking 3 levels was extremely inefficient.