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SOLUTION:
Step 1, have a table to store all the visits sorted by time stamp.
or
have a queue to store the visits per second in the past 5 minute.

Follow-up:
Have two arrays hits[range] and lastupdated[range].
Range = 5 mins = 300 seconds in this case. Any hit within 'range' time from now is valid and should counted.
Index of a hit in the two arrays will be timestamp % 300.
Store the last updated time of a hit. Later if a query for the hit count arrives with a query timestamp, sum up all the valid hits from array 'hits'. Threshold for valid hits: query time - hit last updated time < 300.

public class HitCounter {

    int[] times, hits;
    int timeRange;

    /** Initialize your data structure here. */
    public HitCounter(int range) {
        times = new int[range];
        hits = new int[range];
        timeRange = range;
    }

    /** Record a hit.
     @param timestamp - The current timestamp (in seconds granularity). */
    public void hit(int timestamp) {
        int idx = timestamp % timeRange;
        if (times[idx] != timestamp) {
            times[idx] = timestamp;
            hits[idx] = 1;
        } else {
            ++hits[idx];
        }
    }

    /** Return the number of hits in the past 5 minutes.
     @param timestamp - The current timestamp (in seconds granularity). */
    public int getHits(int timestamp) {
        int res = 0;
        for (int i = 0; i < timeRange; ++i) {
            if (timestamp - times[i] < timeRange) {
                res += hits[i];
            }
        }
        return res;
    }
}

Followup2
For writing,
Concurrency update becomes an issue. Add write lock for protection. But this may slowdown the machine badly.

Move hit counters onto a distributed system. Have several machines counting together. Assign userIDs to diff hosts.
Add LB on top to make sure requests get distributed evenly.
Upon reading, sum up counts across all machines. For a read-heavy system, add cache.

public class CountMethodCounter {

public static int TIME_DURATION = 10;

List<LocalTime> callList = new LinkedList<>();
public void inc(){
SimpleDateFormat sdf = new SimpleDateFormat("hh:MM:ss");
callList.add(LocalTime.now());
}

public void display(){
for(LocalTime localTime:callList){
System.out.println(localTime);
}
}

private boolean isWithinTime(LocalTime l1, LocalTime l2){
if(SECONDS.between(l1,l2)>TIME_DURATION){
return false;
}
return true;
}
public int count(){
int count=0;

LocalTime currentTime = LocalTime.now();
System.out.println("Count() called at ::"+currentTime);
for(LocalTime local:callList){
if(isWithinTime(local,currentTime)){
count=callList.size()-callList.indexOf(local);
break;
}else{
count=-1;
}
}
return count;
}

public static void main(String[] args) throws InterruptedException {
CountMethodCounter countMethodCounter = new CountMethodCounter();
System.out.println("Calling Methoods");
countMethodCounter.inc();
TimeUnit.SECONDS.sleep(10);
countMethodCounter.inc();
TimeUnit.SECONDS.sleep(10);
countMethodCounter.inc();
TimeUnit.SECONDS.sleep(10);
countMethodCounter.inc();
countMethodCounter.inc();
countMethodCounter.inc();
countMethodCounter.inc();
countMethodCounter.display();
System.out.println(countMethodCounter.count());
}
}

public class CountMethodCounter {

    public static int TIME_DURATION = 10;

    List<LocalTime> callList = new LinkedList<>();
    public void inc(){
        SimpleDateFormat sdf = new SimpleDateFormat("hh:MM:ss");
        callList.add(LocalTime.now());
    }

    public void display(){
        for(LocalTime localTime:callList){
            System.out.println(localTime);
        }
    }

    private boolean isWithinTime(LocalTime l1, LocalTime l2){
        if(SECONDS.between(l1,l2)>TIME_DURATION){
            return false;
        }
        return true;
    }
    public int count(){
        int count=0;

        LocalTime currentTime = LocalTime.now();
        System.out.println("Count() called at ::"+currentTime);
        for(LocalTime local:callList){
            if(isWithinTime(local,currentTime)){
              count=callList.size()-callList.indexOf(local);
              break;
            }else{
                count=-1;
            }
        }
        return count;
    }

    public static void main(String[] args) throws InterruptedException {
        CountMethodCounter countMethodCounter = new CountMethodCounter();
        System.out.println("Calling Methoods");
        countMethodCounter.inc();
        TimeUnit.SECONDS.sleep(10);
        countMethodCounter.inc();
        TimeUnit.SECONDS.sleep(10);
        countMethodCounter.inc();
        TimeUnit.SECONDS.sleep(10);
        countMethodCounter.inc();
        countMethodCounter.inc();
        countMethodCounter.inc();
        countMethodCounter.inc();
        countMethodCounter.display();
        System.out.println(countMethodCounter.count());
    }
}

public class CountMethodCounter {

    public static int TIME_DURATION = 10;

    List<LocalTime> callList = new LinkedList<>();
    public void inc(){
        SimpleDateFormat sdf = new SimpleDateFormat("hh:MM:ss");
        callList.add(LocalTime.now());
    }

    public void display(){
        for(LocalTime localTime:callList){
            System.out.println(localTime);
        }
    }

    private boolean isWithinTime(LocalTime l1, LocalTime l2){
        if(SECONDS.between(l1,l2)>TIME_DURATION){
            return false;
        }
        return true;
    }
    public int count(){
        int count=0;

        LocalTime currentTime = LocalTime.now();
        System.out.println("Count() called at ::"+currentTime);
        for(LocalTime local:callList){
            if(isWithinTime(local,currentTime)){
              count=callList.size()-callList.indexOf(local);
              break;
            }else{
                count=-1;
            }
        }
        return count;
    }

    public static void main(String[] args) throws InterruptedException {
        CountMethodCounter countMethodCounter = new CountMethodCounter();
        System.out.println("Calling Methoods");
        countMethodCounter.inc();
        TimeUnit.SECONDS.sleep(10);
        countMethodCounter.inc();
        TimeUnit.SECONDS.sleep(10);
        countMethodCounter.inc();
        TimeUnit.SECONDS.sleep(10);
        countMethodCounter.inc();
        countMethodCounter.inc();
        countMethodCounter.inc();
        countMethodCounter.inc();
        countMethodCounter.display();
        System.out.println(countMethodCounter.count());
    }
}