Calculate a checksum for a string

That's not possible.

If you can't store previous values, it's not possible to create a unique checksum that is smaller than the information in the string.

Update:

The term "reasonably unique" doesn't make sense, either it's unique or it's not.

To get a reasonably low risk of hash collisions, you can use a resonably large hash code.

The MD5 algorithm for example produces a 16 byte hash code. Convert the string to a byte array using some encoding that preserves all characters, for example UTF-8, calculate the hash code using the MD5 class, then convert the hash code byte array into a string using the BitConverter class:

string theString = "asdf";

string hash;
using (System.Security.Cryptography.MD5 md5 = System.Security.Cryptography.MD5.Create()) {
  hash = BitConverter.ToString(
    md5.ComputeHash(Encoding.UTF8.GetBytes(theString))
  ).Replace("-", String.Empty);
}

Console.WriteLine(hash);

Output:

912EC803B2CE49E4A541068D495AB570

You can use cryptographic Hash functions for this. Most of them are available in .Net

For example:

var sha1 = System.Security.Cryptography.SHA1.Create();
byte[] buf = System.Text.Encoding.UTF8.GetBytes("test");
byte[] hash= sha1.ComputeHash(buf, 0, buf.Length);
//var hashstr  = Convert.ToBase64String(hash);
var hashstr = System.BitConverter.ToString(hash).Replace("-", "");