I need to move to the next employee from a struct array.

You also need to move the first textbox insetion out of the - while loop
while (sr.Peek() != -1)
{
//Collects the data abd fills the array of structures.
struc_EmployeeRecord EmpRecord = new struc_EmployeeRecord();

EmpRecord.firstname = sr.ReadLine();
EmpRecord.lastname = sr.ReadLine();
EmpRecord.payrate = sr.ReadLine();

//txtFirstName.Text = EmpRecord.firstname;
//txtLastName.Text = EmpRecord.lastname;
//txtPayRate.Text = EmpRecord.payrate;

EmployeeRecordArray[arrayindex] = EmpRecord;
arrayindex++;

}
txtFirstName.Text = EmployeeRecordArray[0].firstname;//start at first name in array
txtLastName.Text = EmployeeRecordArray[0].lastname;
txtPayRate.Text = EmployeeRecordArray[0].payrate;
}
catch
{
MessageBox.Show("File does not exist.");
}

In Vulpes code you will need add if statement so to ensure not incrementing currentIndex beyond size of the array. If it reaches array max, reset to 0

When you instantiate - EmployeeRecordArray = new struc_EmployeeRecord[26];
To instantiate properly use

 int i = File.ReadAllLines(@"Employee.txt").Length;
int intIndex = i / 3; // count number of names

EmployeeRecordArray = new struc_EmployeeRecord[intIndex];