You can periodically poll a list of the running processes using System.Diagnostics.Process.GetProcesses().
This following code outputs information about starting and exiting processes. Exiting system processes won't be recognized.
class Program
{
    struct ProcessStartTimePair
    {
        public Process Process { get; set; }
        public DateTime StartTime { get; set; }
        public DateTime ExitTime
        {
            get
            {
                return DateTime.Now; // approximate value
            }
        }
        public ProcessStartTimePair(Process p) : this()
        {
            Process = p;
            try
            {
                StartTime = p.StartTime;
            }
            catch (System.ComponentModel.Win32Exception)
            {
                StartTime = DateTime.Now; // approximate value
            }
        }
    }
    static void Main(string[] args)
    {
        List<ProcessStartTimePair> knownProcesses = new List<ProcessStartTimePair>();
        while (true)
        {
            foreach (Process p in Process.GetProcesses())
            {
                if (!knownProcesses.Select(x => x.Process.Id).Contains(p.Id))
                {
                    knownProcesses.Add(new ProcessStartTimePair(p));
                    Console.WriteLine("Detected new process: " + p.ProcessName);
                }
            }
            for (int i = 0; i < knownProcesses.Count; i++) 
            {
                ProcessStartTimePair pair = knownProcesses[i];
                try
                {
                    if (pair.Process.HasExited)
                    {
                        Console.WriteLine(pair.Process.ProcessName + " has exited (alive from {0} to {1}).", pair.StartTime.ToString(), pair.ExitTime.ToString());
                        knownProcesses.Remove(pair);
                        i--; // List was modified, 1 item less
                        // TODO: Store in the info in the database
                    }
                }
                catch (System.ComponentModel.Win32Exception)
                {
                    // Would have to check whether the process still exists in Process.GetProcesses().
                    // The process probably is a system process.
                }
            }
            Console.WriteLine();
            System.Threading.Thread.Sleep(1000);
        }
    }
}
You probably can simply ignore system processes that don't let you read the HasExited property.
Edit
Here's a .net 2.0 version:
    static void Main(string[] args)
    {
        List<ProcessStartTimePair> knownProcesses = new List<ProcessStartTimePair>();
        while (true)
        {
            foreach (Process p in Process.GetProcesses())
            {
                if (!ProcessIsKnown(knownProcesses, p.Id))
                {
                    knownProcesses.Add(new ProcessStartTimePair(p));
                    Console.WriteLine("Detected new process: " + p.ProcessName);
                }
            }
            for (int i = 0; i < knownProcesses.Count; i++) 
            [...]
        }
    }
    static bool ProcessIsKnown(List<ProcessStartTimePair> knownProcesses, int ID)
    {
        foreach (ProcessStartTimePair pstp in knownProcesses)
        {
            if (pstp.Process.Id == ID)
            {
                return true;
            }
        }
        return false;
    }
Note that the code could be improved and only shows the concept.