Note: This solution can not provide 1 Ampere output while boosting from 5 Volts, as pointed out by @markrages. Also, the product mentioned is not suggested to be operated beyond a step-up ratio of 1:3, i.e. up to 15 Volts from a 5 Volt source. Hence, this answer does not meet the criteria specified in the original question. With the amended 12 Volt source, it would work.
As has been pointed out by @Wouter van Ooijen, one is unlikely to find a simple single-IC solution for boosting a 5 Volt DC source to 24 Volts at 1 Ampere.
An alternative that fits the specified budget but not the full requirements is a pre-built adjustable DC-DC boost module such as this one on eBay for $4.49 with free international shipping.
The module uses the Texas Instruments LM2577 step-up voltage regulator, and incorporates the required inductor and adjustment preset on the board.
Input voltage ranges from 3.5 to 30 Volts, and output can be adjusted from 4 to 30 Volts, as long as the output is not set to more than 3 times the input voltage.
Searching on eBay and other sites may yield lower prices for these modules.