having a List of int arrays like:
List intArrList = new List();
intArrList.Add(new int[3] { 0, 0, 0 });
intArrList.Add(new int[5] {         
          List<int[]> CopyString1 = new List<int[]>();
CopyString1.AddRange(intArrList);
List<int[]> CopyString2 = new List<int[]>();
CopyString2.AddRange(intArrList);
for (int i = 0; i < CopyString2.Count(); i++)
{
    for (int j = i; j < CopyString1.Count(); j++)
    {
        if (i != j && CopyString2[i].Count() == CopyString1[j].Count())
        {
            var cnt = 0;
            for (int k = 0; k < CopyString2[i].Count(); k++)
            {
                if (CopyString2[i][k] == CopyString1[j][k])
                    cnt++;
                else
                    break;
            }
            if (cnt == CopyString2[i].Count())
                intArrList.RemoveAt(i);
        }
    }
}
                                                                        You can define your own implementation of IEqualityComparer and use it together with IEnumerable.Distinct:
class MyComparer : IEqualityComparer<int[]> 
{
    public int GetHashCode(int[] instance) { return 0; } // TODO: better HashCode for arrays
    public bool Equals(int[] instance, int[] other)
    {
        if (other == null || instance == null || instance.Length != other.Length) return false;
        return instance.SequenceEqual(other);
    }
}
Now write this to get only distinct values for your list:
var result = intArrList.Distinct(new MyComparer());
However if you want different permutations also you should implement your comparer this way:
public bool Equals(int[] instance, int[] other)
{
    if (ReferenceEquals(instance, other)) return true; // this will return true when both arrays are NULL
    if (other == null || instance == null) return false;
    return instance.All(x => other.Contains(x)) && other.All(x => instance.Contains(x));
}
EDIT: For a better GetashCode-implementation you may have a look at this post as also suggested in @Mick´s answer.
Use GroupBy:
var result = intArrList.GroupBy(c => String.Join(",", c))
                       .Select(c => c.First().ToList()).ToList();
The result:
{0, 0, 0}
{20, 30, 10, 4, 6}
{1, 2, 5}
{12, 22, 54}
{1, 2, 6, 7, 8}
{0, 0, 0, 0}
EDIT: If you want to consider {1,2,3,4} be equal to {2,3,4,1} you need to use OrderBy like this:
var result = intArrList.GroupBy(p => string.Join(", ", p.OrderBy(c => c)))
                       .Select(c => c.First().ToList()).ToList(); 
EDIT2: To help understanding how the LINQ GroupBy solution works consider the following method:
public List<int[]> FindDistinctWithoutLinq(List<int[]> lst)
{
    var dic = new Dictionary<string, int[]>();
    foreach (var item in lst)
    {
        string key = string.Join(",", item.OrderBy(c=>c));
        if (!dic.ContainsKey(key))
        {
            dic.Add(key, item);
        }
    }
    return dic.Values.ToList();
}
                                                                        You can use a HashSet. HashSet is a collection used for guarantee uniqueness and you can compare items on collection, Intersect, Union. etc.
Pros: No duplicates, easy to manipulate groups of data, more efficient Cons: You can't get a specific item in the collection, for example: list[0] doesn't work for HashSets. You can only Enumerating the items. e.g. foreach
Here is an example:
using System;
using System.Collections.Generic;
namespace ConsoleApp2
{
    class Program
    {
        static void Main(string[] args)
        {
            HashSet<HashSet<int>> intArrList = new HashSet<HashSet<int>>(new HashSetIntComparer());
            intArrList.Add(new HashSet<int>(3) { 0, 0, 0 });
            intArrList.Add(new HashSet<int>(5) { 20, 30, 10, 4, 6 });  //this
            intArrList.Add(new HashSet<int>(3) { 1, 2, 5 });
            intArrList.Add(new HashSet<int>(5) { 20, 30, 10, 4, 6 });  //this
            intArrList.Add(new HashSet<int>(3) { 12, 22, 54 });
            intArrList.Add(new HashSet<int>(5) { 1, 2, 6, 7, 8 });
            intArrList.Add(new HashSet<int>(4) { 0, 0, 0, 0 });
            // Checking the output
            foreach (var item in intArrList)
            {
                foreach (var subHasSet in item)
                {
                    Console.Write("{0} ", subHasSet);
                }
                Console.WriteLine();
            }            
            Console.Read();
        }
        private class HashSetIntComparer : IEqualityComparer<HashSet<int>>
        {
            public bool Equals(HashSet<int> x, HashSet<int> y)
            {
                // SetEquals does't set anything. It's a method for compare the contents of the HashSet. 
                // Such a poor name from .Net
                return x.SetEquals(y);
            }
            public int GetHashCode(HashSet<int> obj)
            {
                //TODO: implemente a better HashCode
                return base.GetHashCode();
            }
        }
    }
}
Output:
0
20 30 10 4 6
1 2 5
12 22 54
1 2 6 7 8
Note: Since 0 is repeated several times, HashSet considers the 0 only once. If you need diferentiate between 0 0 0 0 and 0 0 0 then you can replace
HashSet<HashSet<int>> for HashSet<List<int>>and implement a Comparer to the List instead.
You can use this link to learn how to compare a list: https://social.msdn.microsoft.com/Forums/en-US/2ff3016c-bd61-4fec-8f8c-7b6c070123fa/c-compare-two-lists-of-objects?forum=csharplanguage
If you want to learn more about Collections and DataTypes this course is a perfect place to learn it: https://app.pluralsight.com/player?course=csharp-collections&author=simon-robinson&name=csharp-collections-fundamentals-m9-sets&clip=1&mode=live
Using MoreLINQ this can be very simple with DistinctBy.
var result = intArrList.DistinctBy(x => string.Join(",", x));
Similar to the GroupBy answer if you want distinction to be irrespective of order just order in the join.
var result = intArrList.DistinctBy(x => string.Join(",", x.OrderBy(y => y)));
EDIT: This is how it's implemented
public static IEnumerable<TSource> DistinctBy<TSource, TKey>(this IEnumerable<TSource> source,
            Func<TSource, TKey> keySelector, IEqualityComparer<TKey> comparer)
        {
            if (source == null) throw new ArgumentNullException(nameof(source));
            if (keySelector == null) throw new ArgumentNullException(nameof(keySelector));
            return _(); IEnumerable<TSource> _()
            {
                var knownKeys = new HashSet<TKey>(comparer);
                foreach (var element in source)
                {
                    if (knownKeys.Add(keySelector(element)))
                        yield return element;
                }
            }
        }
So you if you don't need MoreLINQ for anything else you can just use a method like this:
private static IEnumerable<int[]> GetUniqueArrays(IEnumerable<int[]> source)
    {
        var knownKeys = new HashSet<string>();
        foreach (var element in source)
        {
            if (knownKeys.Add(string.Join(",", element)))
                yield return element;
        }
    }
                                                                        Well lifting code from here and here. A more generic implementation of GetHashCode would make this more generic, however I believe the implementation below is the most robust
class Program
{
    static void Main(string[] args)
    {
        List<int[]> intArrList = new List<int[]>();
        intArrList.Add(new int[3] { 0, 0, 0 });
        intArrList.Add(new int[5] { 20, 30, 10, 4, 6 });  //this
        intArrList.Add(new int[3] { 1, 2, 5 });
        intArrList.Add(new int[5] { 20, 30, 10, 4, 6 });  //this
        intArrList.Add(new int[3] { 12, 22, 54 });
        intArrList.Add(new int[5] { 1, 2, 6, 7, 8 });
        intArrList.Add(new int[4] { 0, 0, 0, 0 });
        var test = intArrList.Distinct(new IntArrayEqualityComparer());
        Console.WriteLine(test.Count());
        Console.WriteLine(intArrList.Count());
    }
    public class IntArrayEqualityComparer : IEqualityComparer<int[]>
    {
        public bool Equals(int[] x, int[] y)
        {
            return ArraysEqual(x, y);
        }
        public int GetHashCode(int[] obj)
        {
            int hc = obj.Length;
            for (int i = 0; i < obj.Length; ++i)
            {
                hc = unchecked(hc * 17 + obj[i]);
            }
            return hc;
        }
        static bool ArraysEqual<T>(T[] a1, T[] a2)
        {
            if (ReferenceEquals(a1, a2))
                return true;
            if (a1 == null || a2 == null)
                return false;
            if (a1.Length != a2.Length)
                return false;
            EqualityComparer<T> comparer = EqualityComparer<T>.Default;
            for (int i = 0; i < a1.Length; i++)
            {
                if (!comparer.Equals(a1[i], a2[i])) return false;
            }
            return true;
        }
    }
}
Edit: a Generic implementation of IEqualityComparer for an arrays of any type:-
public class ArrayEqualityComparer<T> : IEqualityComparer<T[]>
{
    public bool Equals(T[] x, T[] y)
    {
        if (ReferenceEquals(x, y))
            return true;
        if (x == null || y == null)
            return false;
        if (x.Length != y.Length)
            return false;
        EqualityComparer<T> comparer = EqualityComparer<T>.Default;
        for (int i = 0; i < x.Length; i++)
        {
            if (!comparer.Equals(x[i], y[i])) return false;
        }
        return true;
    }
    public int GetHashCode(T[] obj)
    {
        int hc = obj.Length;
        for (int i = 0; i < obj.Length; ++i)
        {
            hc = unchecked(hc * 17 + obj[i].GetHashCode());
        }
        return hc;
    }
}
Edit2: If ordering of the integers within the arrays doesn't matter I would
var test = intArrList.Select(a => a.OrderBy(e => e).ToArray()).Distinct(comparer).ToList();