If you don't want to group your result, use a window function.
You didn't state your DBMS, but this is ANSI SQL:
SELECT AccountNumber,
Bill,
BillDate,
SUM(Bill) over (partition by accountNumber) as account_total
FROM Table1
order by AccountNumber, BillDate;
Here is an SQLFiddle: http://sqlfiddle.com/#!15/2c35e/1
You can even add a running sum, by adding:
sum(bill) over (partition by account_number order by bill_date) as sum_to_date
which will give you the total up to the current's row date.