Running Total which Resets when Customer changes (keeping existing RowNumbers)- attempt included
This should do what you want:

WITH CTE AS
(
    SELECT  *, 
            RN = ROW_NUMBER() OVER(PARTITION BY Client ORDER BY RowNr)
    FROM #Checks
), CTE2 AS
(
    SELECT  *, 
            RN2 = RowNr - RN 
    FROM CTE 
)
SELECT A.Client, A.RowNr, A.Amount, B.RunTotal
FROM CTE2 A
CROSS APPLY (SELECT SUM(Amount) RunTotal
             FROM CTE2
             WHERE Client = A.Client
             AND RN2 = A.RN2
             AND RowNr &lt;= A.RowNr) B
ORDER BY RowNr


Results:

╔═══════════╦═══════╦════════╦══════════╗
║  Client   ║ RowNr ║ Amount ║ RunTotal ║
╠═══════════╬═══════╬════════╬══════════╣
║ Company A ║     1 ║ 1.00   ║ 1.00     ║
║ Company A ║     2 ║ 1.00   ║ 2.00     ║
║ Company A ║     3 ║ 5.00   ║ 7.00     ║
║ Company B ║     4 ║ 3.00   ║ 3.00     ║
║ Company A ║     5 ║ 1.00   ║ 1.00     ║
║ Company B ║     6 ║ 2.00   ║ 2.00     ║
║ Company B ║     7 ║ 1.00   ║ 3.00     ║
║ Company B ║     8 ║ 5.00   ║ 8.00     ║
╚═══════════╩═══════╩════════╩══════════╝