The easiest way to work with strings in SSIS is via a Script Component.

1. Add a script component transformation
2. For input columns, select the column with the time string. In my example I used time
3. For inputs and outputs, add a column, Seconds and set the datatype to double:
4. On the script tab, hit the edit script button and modify the method, Input0_ProcessInputRow():

   public override void Input0_ProcessInputRow(Input0Buffer Row)
    {
        string current_time = Row.time.Trim();
        try
        {
            if (current_time.StartsWith(":"))
            {
                current_time = "0" + current_time;
            }
            Row.Seconds = TimeSpan.Parse(current_time).TotalSeconds;
        }
        catch (Exception e)
        {
            ComponentMetaData.FireError(10, "Convert to Seconds", e.Message, "", 0, out bool cancel);
        }
    }

Output will look like this: