One thing I like about D&D 5e's character generation is the emphasis it puts on who your character was before they became a level 1 PC. It's a simple enough idea. Your character starts with skills and training, so where did they get those? What was their life like before they decided going into caves, tombs, and dungeons to fight monsters to the death over a bit of coin was the right answer for them?
Other games like L5R kind of have this built into the system. Your character was born a samurai, they were being trained to be an adult samurai, and so that kind of comes with it. Even then though, it is good to think of who the character was before they became an adult.
What were their interests? What did they do for fun? Do they have any embarrassing stories? Any moments they're particularly proud of? Is there a trinket or other small thing they hold onto as a memory?
For that matter, who were their parents? What were they like? How were they raised?
Being able to answer these questions can help make your character have more depth.
When backgrounds are on the table it can be fun to go for strange mixtures of things. What brought a Charlatan to becoming a Paladin? Why is the Acolyte and Assassin? What happened in the war, or after the war, that turned the soldier into a wizard?
You can do this in other settings as well. What happened when the Matsu turned out to be a Shugenja? Why is the Kakita a member of the Daidoji Iron Warriors?
While there's nothing wrong with going the routes that have obvious synergy - it makes sense an Acolyte could then be chosen to become a Cleric and wield divine favor - it can be really interesting to show those changes of life. It also leaves your character with a broader sense of abilities and places they've been, as opposed to a narrow focus and defined area of expertise.
Do Consider Game Size
However, and especially if there are mechanical implications in the background, do consider game size. The more players in a game, the more having a niche can be valuable for claiming your time in the spotlight. Depth more than breadth may be more valuable then. Then again, everyone needs a backup.