So you were on the right track. Inside your componentDidMount() you could have finished the job by implementing setInterval() to trigger the change, but remember the way to update a components state is via setState(), so inside your componentDidMount() you could have done this:

componentDidMount() {
  setInterval(() => {
   this.setState({time: Date.now()})    
  }, 1000)
}

Also, you use Date.now() which works, with the componentDidMount() implementation I offered above, but you will get a long set of nasty numbers updating that is not human readable, but it is technically the time updating every second in milliseconds since January 1, 1970, but we want to make this time readable to how we humans read time, so in addition to learning and implementing setInterval you want to learn about new Date() and toLocaleTimeString() and you would implement it like so:

class TimeComponent extends Component {
  state = { time: new Date().toLocaleTimeString() };
}

componentDidMount() {
  setInterval(() => {
   this.setState({ time: new Date().toLocaleTimeString() })    
  }, 1000)
}

Notice I also removed the constructor() function, you do not necessarily need it, my refactor is 100% equivalent to initializing site with the constructor() function.

Owing to changes in React V16 where componentWillReceiveProps() has been deprecated, this is the methodology that I use for updating a component. Notice that the below example is in Typescript and uses the static getDerivedStateFromProps method to get the initial state and updated state whenever the Props are updated.

    class SomeClass extends React.Component<Props, State> {
  static getDerivedStateFromProps(nextProps: Readonly<Props>): Partial<State> | null {
    return {
      time: nextProps.time
    };
  }

  timerInterval: any;

  componentDidMount() {
    this.timerInterval = setInterval(this.tick.bind(this), 1000);
  }

  tick() {
    this.setState({ time: this.props.time });
  }

  componentWillUnmount() {
    clearInterval(this.timerInterval);
  }

  render() {
    return <div>{this.state.time}</div>;
  }
}

i myself like setTimeout more that setInterval but didn't find a solution in class based component .you could use sth like this in class based components:

class based component and setInterval:

class Clock extends React.Component {
  constructor(props) {
    super(props);
    this.state = {
      date: new Date()
    };
  }

  componentDidMount() {
    this.timerID = setInterval(
      () => this.tick(),
      1000
    );
  }

  componentWillUnmount() {
    clearInterval(this.timerID);
  }

  tick() {
    this.setState({
      date: new Date()
    });
  }

  render() {
    return (
      this.state.date.toLocaleTimeString()
    );
  }
}

ReactDOM.render( 
  <Clock / > ,
  document.getElementById('app')
);

<script src="https://cdnjs.cloudflare.com/ajax/libs/react/16.6.3/umd/react.production.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/react-dom/16.6.3/umd/react-dom.production.min.js"></script>

<div id="app" />

function based component and setInterval:

https://codesandbox.io/s/sweet-diffie-wsu1t?file=/src/index.js

function based component and setTimeout:

https://codesandbox.io/s/youthful-lovelace-xw46p

class ShowDateTime extends React.Component {
   constructor() {
      super();
      this.state = {
        curTime : null
      }
    }
    componentDidMount() {
      setInterval( () => {
        this.setState({
          curTime : new Date().toLocaleString()
        })
      },1000)
    }
   render() {
        return(
          <div>
            <h2>{this.state.curTime}</h2>
          </div>
        );
      }
    }

In the component's componentDidMount lifecycle method, you can set an interval to call a function which updates the state.

 componentDidMount() {
      setInterval(() => this.setState({ time: Date.now()}), 1000)
 }

@Waisky suggested:

You need to use setInterval to trigger the change, but you also need to clear the timer when the component unmounts to prevent it leaving errors and leaking memory:

If you'd like to do the same thing, using Hooks:

const [time, setTime] = useState(Date.now());

useEffect(() => {
  const interval = setInterval(() => setTime(Date.now()), 1000);
  return () => {
    clearInterval(interval);
  };
}, []);

Regarding the comments:

You don't need to pass anything inside []. If you pass time in the brackets, it means run the effect every time the value of time changes, i.e., it invokes a new setInterval every time, time changes, which is not what we're looking for. We want to only invoke setInterval once when the component gets mounted and then setInterval calls setTime(Date.now()) every 1000 seconds. Finally, we invoke clearInterval when the component is unmounted.

Note that the component gets updated, based on how you've used time in it, every time the value of time changes. That has nothing to do with putting time in [] of useEffect.

The following code is a modified example from React.js website.

Original code is available here: https://reactjs.org/#a-simple-component

class Timer extends React.Component {
  constructor(props) {
    super(props);
    this.state = {
      seconds: parseInt(props.startTimeInSeconds, 10) || 0
    };
  }

  tick() {
    this.setState(state => ({
      seconds: state.seconds + 1
    }));
  }

  componentDidMount() {
    this.interval = setInterval(() => this.tick(), 1000);
  }

  componentWillUnmount() {
    clearInterval(this.interval);
  }

  formatTime(secs) {
    let hours   = Math.floor(secs / 3600);
    let minutes = Math.floor(secs / 60) % 60;
    let seconds = secs % 60;
    return [hours, minutes, seconds]
        .map(v => ('' + v).padStart(2, '0'))
        .filter((v,i) => v !== '00' || i > 0)
        .join(':');
  }

  render() {
    return (
      <div>
        Timer: {this.formatTime(this.state.seconds)}
      </div>
    );
  }
}

ReactDOM.render(
  <Timer startTimeInSeconds="300" />,
  document.getElementById('timer-example')
);