You better believe your character has something to hide. Small indiscretions—cheating on a diet, lying to a loved one—are one thing, but big secrets have the ability to completely destroy your character. If they're hiding a shame-based secret, one that is tied to who they are at their core, or something they believe could train-wreck their life (or someone else's), the fear of it coming to light can push them to rearrange their existence around keeping it hidden, resulting in massive fallout in many areas of the character's life.

Being evasive
Withdrawing from others
Claiming that events from the past happened differently than they actually did
Minimizing connections whenever possible: I didn't know him that well, or I don't remember the last time we spoke
Living in denial about what happened
Lying to protect the secret
Changing the subject when certain topics are broached
Deflecting—turning the attention off of themselves and onto someone else
Picking a fight about something else if someone is getting too close to the truth
Having relationships that only go to a certain depth, limiting their intimacy with others
Signs of nervousness or agitation in specific situations
Seeding speculation into the conversation that will lead people away from the truth
Physical ailments, such as insomnia, incontinence, acid reflux, acne breakouts, etc.
Seeking treatment for anxiety, panic attacks, and emotional volatility
Staying busy (to avoid thinking about the secret)
Self-medicating with alcohol or drugs