If I had to guess, I'd make a loop that will reasign the index of each character one step back or something like that but I'm not sure if that's how I should do it.

Yep, that's pretty much it...

Here's how I do it:

char data[10];

if (Serial.available()) {
    char inch = Serial.read();
    for (int i = 0; i < 9; i++) {
        data[i] = data[i+1];
        data[9] = inch;
    }
}

Every incoming character gets added to the end of the array after everything else has been shuffled down one place.

If I had to guess, I'd make a loop that will reasign the index of each character one step back or something like that but I'm not sure if that's how I should do it.

Yep, that's pretty much it...

Here's how I do it:

char data[10];

if (Serial.available()) {
    char inch = Serial.read();
    for (int i = 0; i < 9; i++) {
        data[i] = data[i+1];
    }
    data[9] = inch;
}

Every incoming character gets added to the end of the array after everything else has been shuffled down one place.