Notice: This is only a guide on how to do the achievement's fast and efficiently. If you need an explanation on what an achievement is please visit Cav's guide here.
Eat 100 Food - You get bandages at the start. Open a full inventory of those and eat away.
Kill 20 Chickens - Chickens are located at Lunar Isle. Kill away.
Make 150 Potions - Start by adding eye of newt to Guam potion (unf). Everything you'll need is located in the Herblore shop.
Drink 100 Sips - The potions you just made? Feel free to drink em.
Kill 20 players- This one is a bit of a gamble, but the easiest kills are skillers at ::3xpk
Die 20 times - Again, ::3xpk is your way to go if you wanna die.
Burn 100 Food- You'll easily get this achievement later after the next achievement.
Cook 200 Food - You'll eventually need to fish for food anyways, so its better to do that achievement first and then cook your food after.
Burn 100 of any log - You'll need to woodcut 150 trees for an achievement later, so do that first and then burn all the logs.
Bury 100 bones- Easy achievement to get, bury bones at ::3xpk for the most experience. You can get frost bones from pretty much any store, but dragon bones will work fine too. You'll eventually kill 100 dragons later anyhow. You also get bones from your starter pack (I believe) so those will work too.
Pray 100 bones - Use the bones on the altar south of home.
Woodcut 150 trees - Easy enough, cut 150 trees, burn the logs for experience and achievement.
Mine 100 ores - The best ores to mine would be copper and tin, or iron. Iron is the best here, but depending on your level you may have to mine something else until you can mine iron at level 15.
Smelt 50 bars - Smelt 50 iron ores into iron bars.
Make 50 iron axes - Use the 50 iron bars you made into 50 iron axes.
Fish 50 fishes - You'll eventually need to cook some and fish more anyhow. The best fish to fish are either shark or rocktail, if you have the levels. Otherwise fish anything.
Cut 100 uncuts - You'll easily get this while going for 99 crafting. Simply buy the cheapest gem at the Crafting teleport and cut away.
Unlock 5 Crystal Key chests - Buy 5 crystal keys from players. They are most commonly sold in halves, but you should be able to find 5 keys within 20-30 minutes of simply asking.
(This is where it starts to matter about efficiency)
Kill 100 dragons - There is a hard achievement to kill 25 KBD. Kill 25 KBD first because he counts as a dragon for this achievement, then kill 75 blue dragons in taverly dungeon.
Kill 250 Ghosts - You can get these as a slayer task, but overall it's pretty boring but an easy task if you have good gear. They are located along the path at Taverly dungeon.
Kill Graardor 5 times - This achievement only counts to Graardor himself, so you could be a scumbag and only kill the boss and teleport out.
Kill Kbd 5 times - Easy achievement since you have to kill him 25 times eventually anyhow. Kill Kalphite Queen 5 times - Easy boss, just kill her 5 times.
Kill Barrelchest 5 times - Also easy, it's just 5 kills.
Prestige once - Easy achievement since with 5x exp from dictator, and most skills can be done in 3xpk.
Finish 35 Slayer tasks - This one is a bit more time consuming, but if you are simply looking for fast tasks ask for one, and you can make it an easier task for 5 slayer points.
Fish 350 Sharks - Easy achievement but takes a long time.
Fish 250 Rocktails - Easy achievement, but watch out for pkers at 3xpk.
Woodcut 250 Magic logs - I will get into this explanation later. For now, skip this achievement or at least don't grind cut 250 logs.
Obtain 150 PC Points - The rewards are pretty nice here, and you'll have to get 600 for all PC achievements. Points given are 8-20, but average at 8-13 unless you have OP gear.
Obtain 150 PK points - Kill skillers at ::3xpk during the PK event for 5 extra PK points.
Unlock 10 Crystal key chests - Easy to buy keys or halves from players. You'll need 25 total keys for every chest achievement.
Brew 50 of all extreme pots - N/A
Brew 75 Overloads - N/A
Clean 500 herbs - Easy early herblore levels you can clean herbs. Get them from farming.
Steal from stalls 500 times. - You'll easily get this while stealing 100m GP from stalls.
Mine 250 runite ores - Easy achievement, you'll need the ores for the next one.
Smith 50 Rune Platebodies - Use the ores to smith the platebodies until you get the achievement.
Pray 625 bones - The only way you are gonna get this achievement is by grinding dragons or another npc, collecting the bones, and praying them at the altar unless you buy them.
Catch 25 Magpie implings - Easy achievement especially because they give really good hunter xp.
Hard & Legendary
(Due to most of the tasks being repetitive I decided to group the two sections together and only mention the ones that need mentioning. If you need advice on any of the other achievements not listed, ask in this thread and I will answer or look in previous sections.)
Kill 2000 Monsters - Best monsters to kill for this would be Ghosts, Rock crabs, Dark beasts, tormented demons, or abyssal demons since you need to kill a chunk of those for achievements.
Obtain a 1B Ticket - Once you get 1b cash, you can turn it into a ticket. Best way to get cash is kill Penance queen or ::tarn, but you can get it from other bosses or skilling. If you turn it from a ticket, back into cash, and back into a ticket again, it will count towards the "Obtain a 1b ticket 5 times" achievement.
Unlock 50 Barrows Chests - I am not sure how to do this achievement, or if it is even possible anymore. Woodcut/Burn X Magic Logs - There are various achievements regarding cutting and burning magic logs. The best way I have found is to use the inferno adze because it counts towards burning and cutting a log. If you use an axe, such as a rune axe, and keep the adze in your inventory it still incinerates them. Wielding a rune axe and adze in inventory, it's a 99.9% chance it incinerates the log (from my experience.)
Forge a dragonfire shield - Kill dragons for the visage, or swap someone 50m or a dfs for it.
Assemble a god sword - Buy the pieces and a hilt, or get some from drops. Bandos is probably the easiest.
Vote 200 times - This counts to only redeeming the code, not trading the vote ticket to other players.
Make a divine spirit shield - Get the following items - Spirit Shield, Holy Elixer and Divine Sigil. Use elixer on the shield to bless it, then use the sigil on the blessed spirit shield.
Smith 1500 Rune Platebodies - This achievement is currently bugged so you only have to use 1500 bars. 300 platebodies. Easiest way is to buy bars from the summoning shop.
High Alchemy items with a total value of 1B GP - Best items to alch are the drops from corp (like tuna potatos) or skilling items like rocktail.
Win 75M from Gambler - DO NOT JUST GO AND THROW 5M OUT EVERY TIME. The Gambler is awkward, and for some reason you seem to lose more the higher up your bets go. I have lost 300m to this achievement and only have 10m towards it. It's better to play the Flower Game too, at least in my experience.
(I will update with more pictures and information later to come. I just wanted to share my experiences with it so far.)