To "unlock" a relic you need to complete four of the eight quests associated with it. As with everything in Bejweled, luck plays a big part. Sometimes everything will drop perfectly and you will beat the quest with ease, and sometime you will fail badly.
Butterflies isn't timed so take your time and plan your moves. If a butterfly gets near the top and you can't kill it, try to remove some jewels underneath it so it drops back down. Sometimes the butterflies kill themselves by moving into same-coloured jewels, so try to look ahead and don't waste moves on butterflies who are going to die anyway.
Gold Rush requires a bit of luck to complete. If you get stuck with matches only appearing near the top of the board, restart. Ideally you want to make some special jewels at the bottom to clear the dirt away fast.
For Alchemy I found it easiest to concentrate on completing horizontal lines. Try to complete the bottom lines at the start of the game so you aren't stuck with no matches there later.
Stratamax is quite easy. Just aim to make special jewels as often as you can and you will probably beat each one on your first go.
Balance requires you to... balance! Try to avoid big combos because you may get a huge number of jewels of one type which you won't be able to recover from. Sometimes it's good to just sit for a moment and wait for the weights to move to a safer position, if they get too far apart.
Avalanche is very similar to Stratamax. The same tactic applies; concentrate on making special jewels.
For Buried Treasure and Sandstorm, try to concentrate on one treasure at a time. This is especially important on the harder stages where the sand comes back if you take too long.
Time Bomb sounds stressful but it really isn't. Personally, I only found Timb Bomb IV to be difficult, as the bombs have 45 second timers. After that the 60 second timers on Time Bomb V feel like a luxury, and you shouldn't have much trouble removing the bombs before the timer runs out.
Poker is a bit luck based. Try to work out which colour has the most matches before you start the hand, and then match the ones at the top of the board first so the don't mess up the ones lower down. With a bit of luck another match of the same colour will drop. If you can only make a pair, try to make a pair of a different colour afterwards. Hopefully some jewels that match one of your pairs will drop and you can get a Full House. Getting Flush and Full House requires good luck with Hypercubes. Try to make Hypercubes whenever possible. They count as wild cards in the current hand, and make it easy to get the colour you want in the future because you only have to match one jewel with them.
For Ice Storm you need to match fast and then you won't have many problems. If some ice reaches the top, don't panic, just work on removing it before it solidifies. Vertical matches completely remove the ice column, but horizontal matches are useful when you're in trouble so don't neglect them.
Wall Blast also requires a bit of luck. Concentrate on making special jewels and manoeuvring them into the right spot before detonating them. Special jewels also drop onto the board from time to time so make good use of them.
After you complete four quests at the fifth relic, the achievement will unlock.