If you convert each character to a string first, then use int.Parse (or still Convert.ToInt32) that will work.
Personally I'd use LINQ for this, e.g.
int[] sequence = array.Select(x => int.Parse(x.ToString())).ToArray();
... or use ToList if you're just as happy with List<int>.
If you want to use Char.GetNumericValue as suggested in another answer, you can use that with LINQ too:
int[] sequence = array.Select(x => (int) char.GetNumericValue(x)).ToArray();
Note that the cast to int is required because char.GetNumericValue returns a double. (There are Unicode characters for values such as "a half" for example.)
Or if you're absolutely sure you're just going to have ASCII digits, you could use the quick and dirty:
int[] sequence = array.Select(x => x - '0').ToArray();
But I get the ASCII coding of 1,2,3,4 not the numbers themselves. Actually, you get the 32-bit signed integer of the Unicode character.