Your input reading part is basically correct. However, I'd rewrite it using better array indexes, and verifying that the scanning worked fine, and that the score is within acceptable range. For example:
Code:
#include <stdlib.h>
#include <stdio.h>

#define NUM_MOVIES  5
#define NUM_SCORES_PER_MOVIE 5

int main(void)
{
    int score[NUM_MOVIES][NUM_SCORES_PER_MOVIE];
    int m; /* Movie index */
    int s; /* Score index */

    for (m = 0; m < NUM_MOVIES; m++) {
        for (s = 0; s < NUM_SCORES_PER_MOVIE; s++) {
            if (scanf("%d", &(score[m][s])) != 1 ||
                score[m][s] < 1 || score[m][s] > 10) {
                fprintf(stderr, "Invalid score %d of %d for movie %d of %d.\n",
                                s+1, NUM_SCORES_PER_MOVIE, m+1, NUM_MOVIES);
                return EXIT_FAILURE;
            }
        }
    }

    /* TODO */

    return EXIT_FAILURE;
}
The +1 in the movie and score numbers in the error messages are there, because humans start counting at 1, and C at 0.

Make sure the above works, maybe by adding temporarily a second double loop to print each score given for each movie. Compile it, with all warnings enabled in your compiler, and work on it until it compiles without warnings, and works when you give it the input. (That is, if you give it incorrect input, it complains about it.)

Only then worry about how to handle the rest. One step at a time. Make sure it is solid, before relying on it. If you're halfway across the bridge, and it collapses beneath you, it's one heck of a job to find out the cause. But if you proceed only one step at a time, you almost always know where the failure occurred. Without falling down into the chasm.

For the second part, you need a loop over each movie. Inside that loop, you must calculate the average score for the movie, and find the maximum score; that means a loop over the scores. It also means you probably want two arrays (each of NUM_MOVIES elements): one for the average, and one for the maximum.

For the third part, you print out the statistics (average and maximum scores) for each movie.

You could finally reveal which movie got the highest maximum score, and which movie got the highest average score.