// I have everything working except for the dead cell rule
import java.util.Arrays;
import java.util.Random;
public class GOL {
   public static void main(String[] args)
   {
      Cell   newcell   = new Cell(0);
      int    board[][] = new int[5][5];
      Random rand      = new Random();

//https://stackoverflow.com/questions/43314281/random-matrix-fill-between-1-and-0-in-java
      for (int i = 0; i < 5; i++)
      {
         for (int j = 0; j < 5; j++)
         {
            // you could maybe do a random initialization of board before you start checking for the cells
            // in that way it will be more obvious where the problem is defined and when you start to find the solution
            board[i][j] = rand.nextInt(2);
            try{
               // don't use this object(newcell) just in order to be able to call method of it's class,
               // it would be better to make that method static so Cell.getNumberofNeighbors(),
               // look on when to use static methods,
               // or if you make class Board that consists of Cell's you could write it like this
               // board.getNumberOfNeighborsForCurrentCell(cell. or i,j whatever),
               // if you do it like this there is no need to pass the whole matrix everytime,
               // but even then there is space for improvement in class Board make method isDying(cell),
               // add your methods for every rule, that will improve readability, it will be more like a story...
               if (newcell.getNumberofNeighbors(i, j, board) < 2)
               {
                  board[i][j] = 0;
               }
               if (newcell.getNumberofNeighbors(i, j, board) > 3)
               {
                  board[i][j] = 0;
               }
            }
            catch (Exception e) {
               //https://stackoverflow.com/questions/409784/whats-the-simplest-way-to-print-a-java-array
               System.out.println(Arrays.deepToString(board).replace("], " + "[", "\n"));
            }
         }
      }
   }
}


import java.util.Arrays;
import java.util.Random;
public class Cell {
   int numberofneighbors;
   public Cell(int numberofneighbors)
   {
      this.numberofneighbors = numberofneighbors;
   }

   // optimize your parameters, naming is not good, third arg. is really not position
   // it is the whole board(your matrix), so this is rather confusing
   public int getNumberofNeighbors(int x, int y, int[][] position) throws Exception
   {
      if ((x < 0) || (y < 0) || (x > 5) || (y > 5)) { throw new Exception(""); }
      else
      {
         if (position[x][y] == 1)
         {
            if (position[x][y + 1] == 1) { numberofneighbors += 1; }
            if (position[x + 1][y] == 1) { numberofneighbors += 1; }
            if (position[x + 1][y + 1] == 1) { numberofneighbors += 1; }
            if (position[x - 1][y] == 1) { numberofneighbors += 1; }
            if (position[x][y - 1] == 1) { numberofneighbors += 1; }
            if (position[x + 1][y - 1] == 1) { numberofneighbors += 1; }
            if (position[x - 1][y + 1] == 1) { numberofneighbors += 1; }
         }
      }
      return numberofneighbors;
   }
}