In plugin settings, under Integration->WooCommerce you can turn on a plugin option that will add customer data into the data layer. It will populate new data layer variables for you:
{
"customerTotalOrders": 0,
"customerTotalOrderValue": "0.00",
"customerFirstName": "",
"customerLastName": "",
"customerBillingFirstName": "",
"customerBillingLastName": "",
"customerBillingCompany": "",
"customerBillingAddress1": "",
"customerBillingAddress2": "",
"customerBillingCity": "",
"customerBillingPostcode": "",
"customerBillingCountry": "",
"customerBillingEmail": "",
"customerBillingPhone": "",
"customerShippingFirstName": "",
"customerShippingLastName": "",
"customerShippingCompany": "",
"customerShippingAddress1": "",
"customerShippingAddress2": "",
"customerShippingCity": "",
"customerShippingPostcode": "",
"customerShippingCountry": "",
}
These will be only populated with any value if there is a logged in user on your site.