Initialize array only for test purpose:

int dim = 5;
char ch="A";
String[][] array = new String[dim][];
for( int i = 0 ; i < dim ; i++ ) {
    array[i] = new String[dim];
    for( int j = 0 ; j < dim ; j++, ch++ ) {
        array[i][j] = "" + ch;
    }
}

Output our matrix:

for( int i = 0 ; i < dim ; i++ ) {
    for( int j = 0 ; j < dim ; j++, ch++ ) {
        System.out.print( array[i][j] + " " );
    }
    System.out.println();
}
System.out.println( "============================" );

Solution

Element indexes from diagonals have one rule – their sum is constant on one diagonal:

Variant 1

Use two loops to extract all diagonals.

First loop extracts top half of diagonals:

for( int k = 0 ; k < dim ; k++ ) {
    for( int j = 0 ; j <= k ; j++ ) {
        int i = k - j;
        System.out.print( array[i][j] + " " );
    }
    System.out.println();
}

Second loop iterates on bottom half of diagonals:

for( int k = dim - 2 ; k >= 0 ; k-- ) {
    for( int j = 0 ; j <= k ; j++ ) {
        int i = k - j;
        System.out.print( array[dim - j - 1][dim - i - 1] + " " );
    }
    System.out.println();
}

Variant 2

Use one loop to extract all diagonals, but there are extra iterations and one additional check:

for( int k = 0 ; k < dim * 2 ; k++ ) {
    for( int j = 0 ; j <= k ; j++ ) {
        int i = k - j;
        if( i < dim && j < dim ) {
            System.out.print( array[i][j] + " " );
        }
    }
    System.out.println();
}

The output:

A B C D E 
F G H I J 
K L M N O 
P Q R S T 
U V W X Y 
============================
A 
F B 
K G C 
P L H D 
U Q M I E 
V R N J 
W S O 
X T 
Y 

Update

There are questions about rectangular matrix (height != width) in comments. Here is solution for rectangular matrix:

The rule remains the same: Sum of element indexes from the same diagonal is constant

The minimum sum of indexes is 0 (for first element in matrix with indexes [0;0])

The maximum sum of indexes is width + height – 2 (for last element in matrix with indexes [height-1; with-1])

Initialize rectangular matrix only for test purpose:

int WIDTH = 7;
int HEIGHT = 3;
char ch="A";
String[][] array = new String[HEIGHT][];
for( int i = 0 ; i < HEIGHT ; i++ ) {
    array[i] = new String[WIDTH];
    for( int j = 0 ; j < WIDTH ; j++, ch++ ) {
        array[i][j] = "" + ch;
    }
}

Print our rectangular matrix:

for( int i = 0 ; i < HEIGHT ; i++ ) {
    for( int j = 0 ; j < WIDTH ; j++, ch++ ) {
        System.out.print( array[i][j] + " " );
    }
    System.out.println();
}
System.out.println( "============================" );

Solution

for( int k = 0 ; k <= WIDTH + HEIGHT - 2; k++ ) {
    for( int j = 0 ; j <= k ; j++ ) {
        int i = k - j;
        if( i < HEIGHT && j < WIDTH ) {
            System.out.print( array[i][j] + " " );
        }
    }
    System.out.println();
}

Output:

A B C D E F G 
H I J K L M N 
O P Q R S T U 
============================
A 
H B 
O I C 
P J D 
Q K E 
R L F 
S M G 
T N 
U