Crash Bandicoot N. Sane Trilogy

It's much easier to get this achievement and all the relics if you wait until you have defeated N. Cortex after acquiring 25 crystals. Defeating him unlocks the 'Sprint' ability making all the relic times (on dry land at least) easier.

After that it's a case of picking 5 easy level to sprint through, I found the Medieval levels (Toad Village, Gee Wiz and Double Header) to all be very easy using this method.

Midnight run, Orient Express, and Dino Might are also pretty easy to do using this method.
Always be on the lookout for the yellow 'Timer Freezing' boxes too, stopping the clock for 1, 2 or 3 seconds depending on the number written on the side.

You can usually beat the times very easily without hitting all the boxes though so don't waste time doubling back for them or re-trying for a perfect run, you only need blue relic times for this achievement.