Specifically related to 5E, can I use the Expertise ability granted by Rogue class ability to increase my proficiency in a skill that is granted by racial or background? Specifically, I am making a Criminal (Enforcer) Half-orc Rogue. Can I use Expertise to double the proficiency bonus for those skills if I didn't take the proficiency with the class?
The rogue's expertise reads
At 1st level, choose two of your skill proficiencies, or one of your skill proficiencies and your proficiency with Thieves' Tools. Your proficiency bonus is doubled for any ability check you make that uses either of the chosen proficiencies. At 6th level, you can choose two more of your proficiencies (in skills or with thieves' tools) to gain this benefit.
It doesn't specify that it has to be obtained from the class, so you can pick any of your skills that you are proficient with. Do note however, that the only tool proficiency you can apply it to is with thieves tools.
Rules as Written
There's actually a weird quirk that I discovered while looking into this (which caused me to Google the bug and find this thread)
According to the Player's Handbook pages 11-13 (Chapter 1: Step - by - Step Characters):
Once you have a character in mind, follow these steps in order
PHB 11 (emphasis mine)
Then the steps:
1. Choose a Race
2. Choose a Class
... On your character sheet, record all the features that your class gives you at 1st level.
3. Determine Ability Scores
4. Describe Your Character
... A background gives your character a background feature (a general benefit) and proficiency in two skills, and it might also give you additional languages or proficiency with certain kinds of tools. Record this information, along with the personality information you develop, on your character sheet.
PBH 11-13 (summarized)
Following these steps in order (as directed), you will record your class feature (including expertise) before you have selected a background. Therefore at level 1 a Rogue can apply expertise to their racial skills, but not to their background skills
Rules as Interpreted
Reading the chapter on Backgrounds:
Every story has a beginning. Your character’s background reveals where you came from, how you became an adventurer, and your place in the world. Your fighter might have been a courageous knight or a grizzled soldier. Your wizard could have been a sage or an artisan. Your rogue might have gotten by as a guild thief or commanded audiences as a jester. Choosing a background provides you with important story cues about your character’s identity. The most important question to ask about your background is what changed? Why did you stop doing whatever your background describes and start adventuring? Where did you get the money to purchase your starting gear, or, if you come from a wealthy background, why don't you have more money? How did you learn the skills of your class? What sets you apart from ordinary people who share your background?
PHB 125 (emphasis mine)
It's written as though your background takes place chronologically before level 1. You're effectively "level 0" during this part of your life, and when you give up whatever you were doing before is when you take your first level in an adventuring class. This suggests you should pick a background before you pick a class - or at the very least that any skills or features of your background should apply at the time of initial level up.
It also seems odd that Rogue's two most signature skills (stealth and sleight of hand) would be barred from gaining expertise unless a Rogue foregoes the two backgrounds that sounds like they were made for Rogue: Charlatan and Criminal.
Taking these two facts into account, it seems reasonable to assume that background skills can also receive expertise.
When in doubt, ask your DM. DMs have the final authority on how they choose to interpret the rules.