Just bought? Your purchase price is the best evidence you will ever have
A recent arm's-length sale of the subject property is the strongest single item in an assessment appeal.
Boards weigh evidence by how directly it speaks to the property. Nothing speaks more directly than what the property itself just sold for.
The comparison to run
Take your assessed value, divide by 0.3333 (outside Cook). That is the county's implied opinion of what your house would sell for. Compare it to what you actually paid. If the implied value is meaningfully above your price, you have a straightforward market-value case.
What "arm's length" means
A sale between unrelated parties, marketed openly, neither side under pressure. That is what counts. Sales that generally do not: family transfers, foreclosures and REO, short sales, estate sales under time pressure, and portfolio transactions.
Timing
The closer to the assessment's valuation date, the better. A sale two months before is powerful; four years before, much less so.
What to bring
The closing statement or deed showing price and date, and the listing history showing it was openly marketed. If the sale included personal property or seller credits, be ready to explain the adjustment — an assessor will find it.
The catch
If you paid more than the implied value, do not file on market value. You would be handing the assessor evidence to raise you. Look at uniformity instead, which does not depend on price.