Viola Davis: My 'No. 1 Fear' Is That My Daughter Will Grow Up Feeling Entitled

"I wanted my life to mean and be something deeper," Oscar winner Viola Davis tells PEOPLE in this week's issue about the moment she wanted to become a mom

With a fresh Oscar in hand, a hit television show and no signs of slowing down anytime soon, Viola Davis has a lot going for her career at the moment. But there was a time in her life when her current level of success was another universe of possibility, considering she experienced extreme poverty as a child.

When asked how she handles the balance of wanting her daughter Genesis to have more than Davis did growing up and appreciating it as it comes, the actress admits it’s a never-ending challenge.

“You just listed my number one fear, which is entitlement,” Davis, 51, says in this week’s PEOPLE cover story of keeping her 6-year-old grounded. “I never had a house; Genesis has a house. I do shop at Target, I buy all her clothes at Target or H&M. And maybe, if I’m feeling really good, Nordstrom Rack.”

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Davis — who nabbed her brand-new Academy Award for her role in the Denzel Washington-led drama Fences — says one key is that she and husband Julius Tennon, a fellow actor whom she has been married to since 2003, hold their daughter “accountable for everything,” even at her young age.

Watch the full episode of People Cover Story: Viola Davis streaming now on People/Entertainment Weekly Network (PEN). Go to PEOPLE.com/PEN, or download the app on your favorite device.

“Julius is really tough. He has two beautiful children and seven grandkids,” says the How to Get Away with Murder star. “I came into a relationship where he already had children and grandchildren and raised his kids on his own, so he’s tough — he toes the line, but in a very loving way.”

Continues Davis, “He holds her accountable. Me, not so much — I’m the softie. Really, entitlement. Listen, there are poor kids who are entitled. So I pray.”

RELATED VIDEO: Viola Davis Opens Up About Her Parents and Her Impoverished Childhood

The mom of one — who says “spending the day at Chuck E. Cheese’s” is something simple she loves to do with her daughter — shares that she has “been taking care of babies” since she was 5 or 6 years old and was used to being around kids, but wasn’t sure until she found career success that she wanted to expand her own family.

“You know what it was? I always just wanted to be different. I didn’t want to live a provincial life. In my brain I thought if I got married and had a kid, that’s just too ordinary, and I wanted to be extraordinary,” she explains.

“It was great, but it was not fulfilling,” Davis continues of the “extraordinary life” that did end up happening for her. “I wanted my life to mean and be something deeper. That’s when the urge came, and the urge was just very, very, very, strong. So I have a kid now and she is just the light of our lives.”

For more about Viola Davis’ family life and her history-making Oscar win, pick up the latest issue of PEOPLE, on newsstands now.