Rounding out Your Character
Once you've selected and recorded all of the proficiency, class features, spells and equipment for your character it's a good idea to spend some time to flesh out their background and lore.
Background
In 5e, characters select from a host of backgrounds. This provides the character with some equipment, a story-esque feature, and some lore and guidlines for their backstory. In addition, each background provides several example character traits to help flesh out your character and backstory. You may select or role on these additional tables as you wish. You may also use them or tables from other backgrounds to get inspiration for your character.
Ideal and Bond
These two tables in your background can get you inspiration for background related motivations of your character.
Alignment
I've never been a fan of the traditional D&D alignment system. It's rigid, encourages poor roleplay and unrealistic.
Instead we will use a more fluid system I believe I first heard from Dael Kingsmill.
There are five spectrum of thought, belief and ideals that you will determine where your character belongs. They are:
Reason vs Passion
Authority vs Freedom
Self vs Others
Deliberate vs Spontaneous
Tradition vs Innovation
Pick at least 2-3 of these and determine where along that spectrum your character is. Of course, feel free to decide where they are along all five, but that can be figured out over the course of play as well.
Essential Character Questions
In order to help you get into the headspace of your character, answer the following questions.
Who are your family?
What do you fear?
What do you need that you don’t have?
How do you solve your problems?
How do you create them?
When did you fail?
Try to have an answer to each of these questions. It can be as simple or detailed as you desire. These questions will also help the GM craft a story around your character and provide roleplay opportunities. However, don't get hung up on this or any other details on this page. It's perfect fine to start play not really knowing your character, developing them over time through play.
Description
You should imagine what your character looks like.
What does your character look like?
Are they typical height and weight for their race?
What sort of of clothes do they wear?
Skin, hair, eye color?
How do they present themselves to the public?
Character Bond
Now you need to come up with a link to another player character. Work with at least one other player to come up with a brief one or two sentence describing an adventure, bond, debt, or rivalry your characters share. Are you related? Do you like each other? How did you meet? Etc....
If you'd like, take a look at the phase trio from FATE Core for inspiration. Take note that aspects are not a part of D&D.
Continue to Flesh out As Desired
If you wish, you may continue to answer questions to flesh out your character's backstory. However this is not required and sometimes it's fun to learn who your character is through play.
There are tons of questionnaires and guidelines online for fleshing out a character, below are some ideas:
Why did you become an adventurer?
What motivates you?
How were you trained?
What is your greatest flaw?
What is your biggest secret?
What social class are you from?
What does your character think about religion?
What does your character think about your alliance? The other alliances?
Who are your enemies?
How old is your character?
What is your character's hobby?
What traits do they look for in a friend?
Describe one sentimental object your character keeps with them
Who do you admire?
Who do you hate?
What is your pet peeve?
What are your long term goals?
How does your character want to die?