Update
As of React v16.6.0, you can use the context API like:
class App extends React.Component {
    componentDidMount() {
       console.log(this.context);
    }
    render() {
       // render part here
       // use context with this.context
    }
}
App.contextType = CustomContext
However, the component can only access a single context. In order to use multiple context values, use the render prop pattern. More about Class.contextType.
If you are using the experimental public class fields syntax, you can use a static class field to initialize your contextType:
class MyClass extends React.Component {
  static contextType = MyContext;
  render() {
    let value = this.context;
    /* render something based on the value */
  }
}
Render Prop Pattern
When what I understand from the question, to use context inside your component but outside of the render, create a HOC to wrap the component:
const WithContext = (Component) => {
  return (props) => (
      <CustomContext.Consumer>
           {value =>  <Component {...props} value={value} />}
      </CustomContext.Consumer>
  )
}
and then use it:
class App extends React.Component {
    componentDidMount() {
       console.log(this.props.value);
    }
    render() {
       // render part here
    }
}
export default WithContext(App);