Yes, it's a perfectly fine sentence in English, especially conversational English. For formal writing you'd want to use the possessive form, "your"--I'm tired of your blaming me for everything--because blaming is a gerund, and the possessive form of the pronoun should technically go before a gerund. Another example: "I disapprove of his smoking" is correct, but you might hear English speakers use the objective case for the pronoun, "I disapprove of him smoking."

of blaming myself