How to calculate sum of amount for each 5 weeks before in linq?

You can use the methods in EntityFunctions to perform date and time arithmetic. So you should start by working out the start and end dates, then use TruncateTime if necessary to truncate your created date to a date (instead of date and time), and use DiffDays to work out "number of days since the start of the period". Then just divide by 7 to group...

DateTime today = DateTime.Today;
DateTime start = today.AddDays(-(int) today.DayOfWeek) // Sunday...
                      .AddDays(-28);                   // 4 weeks ago
DateTime end = start.AddDays(7 * 5);

var result = from entry in db.Entries
             where entry.Created >= start && entry.Created < end
             group entry.Amount by EntityFunctions.DiffDays(start, 
                 EntityFunctions.TruncateTime(entry.Created)) / 7 into g
             select new { Week = g.Key + 1, Sum = g.Sum() };

While I'd expect that to work, I haven't personally done any date/time work in EF myself. The general approach should be fine, it's just that you may need to tweak it. Also note that this won't give you any results for weeks that don't have any entries - it's probably easiest to do that outside EF.

EDIT: If the summing part isn't working, it's easy to do the summing locally instead:

var query = from entry in db.Entries
             where entry.Created >= start && entry.Created < end
             group entry.Amount by EntityFunctions.DiffDays(start, 
                 EntityFunctions.TruncateTime(entry.Created)) / 7;

var result = query.AsEnumerable() // Execute the rest locally
                  .Select(g => new { Week = g.Key + 1, Sum = g.Sum() });