You've asked two questions in your post: one on the order of the steps of character creation, and one on which skill proficiencies you can gain from your background.
For the order of application: Character creation is broken into steps for ease of creation, not out of necessity. Whether you start with the background, race, or class is irrelevant since the process is a whole not a stepping program.
The only exception to this was the one I left out, which is ability score rolls. I assume standard array when dealing with questions unless OP states otherwise; however, I'll address that in the event your group is playing with a more hardcore mindset and has you roll ability scores after the character has been built, you may end up with a rather weak Barbarian, an incredibly irritating Bard (so a Bard), or a stupid Wizard. Of course, you could end up with an extremely dexterous warlock or a very durable sorcerer!
The point here is that if your group has different rules in play than the PHB does, it will significantly alter the answer. As per RAW, the order is entirely irrelevant.
For the skill proficiencies: You can do whatever you want for each proficiency of your background.
Your answer is under the rules for customizing a background (PHB, pg. 126; emphasis mine):
You might want to tweak some of the features of a background so it
better fits your character or the campaign setting. To customize a
background, you can replace one feature with any other one, choose
any two skills, and choose a total of two tool proficiencies or
languages from the sample backgrounds. You can either use the
equipment package from your background or spend coin on gear as
described in chapter 5. (If you spend coin, you can’t also take the
equipment package suggested for your class.) Finally, choose two
personality traits, one ideal, one bond, and one flaw. If you can’t
find a feature that matches your desired background, work with your DM
to create one.