If I owe both CRA and a private creditor, does CRA's claim get paid first?

Not always, and this is genuinely fact-dependent rather than a simple always-first-in-line rule. CRA has strong priority for certain specific categories of tax debt, particularly amounts an employer or business held in trust for the government, like unremitted source deductions and collected-but-unremitted GST/HST, reflecting the idea that this money was never really the debtor's own to begin with.

For ordinary income tax debt, though, CRA's priority relative to other creditors can depend on the type of debt involved and on what steps CRA has actually taken, such as registering a certificate against your property, which can affect where CRA's claim ranks compared to other secured or unsecured creditors. This means two people who each owe CRA and a private creditor can end up in genuinely different priority positions, depending on the specific debts involved and the collection steps taken on each side. If you're dealing with competing debts to CRA and a private creditor, working out where each claim actually stands, rather than assuming CRA is simply first, or simply treated like any other creditor, depends on your specific facts, and is worth sorting out carefully rather than guessing.