Translation:I will blame no one and demand nothing.
Are you sure 'claim' was the reason it was marked wrong? Your translation is awkward in other ways, because it's unclear whether the "not" applies to all of it, or just the blaming. A better translation is the one given: "I will blame no one and demand nothing", then you don't have the strange partial negation (does it mean you will NOT claim nothing - therefore claim something?)
"Claim" and "demand" are very similar, but I think the above posts explore the differences quite well. "Claim" tends to imply that there is some (valid?) basis for the claim - even if only in the mind of the claimant. It is also slightly more formal or official sounding. "Demand", on the other hand, need not have a valid basis. A child can demand sweets, or a bank robber demand money - in neither case are they appealing to justice or fairness - it just means: "I want what I want!"