These achievements are all best attempted in a single playthrough of Endless Run mode (Canyon level is recommended) because you will need a significant amount of continues to be able to progress far enough to complete each of these achievement's requirements. The exact amount will depend on your skill (how often you die), but also on luck as the levels are randomly generated (including the items you need for the achievements). The farther the distance you run, the faster your character will move, increasing the difficulty and the need for continues as you progress. I would say having around 60 continues (total cost of 150,000 coins/150 diamonds) is a good amount to attempt this with, but having more is always better. In the Store you can purchase group of continues which will give you the most for your money (5, 10, or 20 continues). Make sure you are quick when you die, as you only have a second or two to hit the "Save Life" button to continue. If you miss it, your run will end and you will have to start over. It is best to practice several times to get a feel for this mode. Once you have enough continues and are ready, attempt your run.
Running 25000 meters is probably the easiest one and will come first. Many of the other achievements will require you run over 100,000 meters.
Collecting 10 power-ups is also quite easy as long as you have them all unlocked (by purchasing at least the first level of each of them in the Store with coins/diamonds).
Collecting 2500 coins is also one of the easier ones.
Destructible targets includes both enemies that pop-up as well as stationary barriers, so be sure to shoot everything. Chests may even count as well. Destroying 35 will take a good while.
Chests are "collected" by shooting them. Getting 8 could potentially take the longest of any as they do not appear that frequently.
Turns are bends in the road. Forks in the road where you choose a direction should also count, but this is unclear. 60 of these will take a while as well.
The problem is none of the achievements unlock until you die and don't use a continue (finish the run). Use your best judgement -- unless you are certain you have met all 6 of the requirements for the achievements, you should keep going until you run out of continues.
The only achievement you can see your active progress towards while in-game is the coins achievement, which shows up in the top left corner. The score that is shown above the coins is related to your distance, but it is not equal and is in fact more than your distance. The other achievements display no real counters at all, so you also need to roughly keep track mentally how many you have. If you finish and do not meet the requirements for any of the achievements, you can check the badges (which have semi-useful counters next to them) at the finishing screen on the right-hand side.
If you fail to get them all on your first attempt, you will need to earn more coins to purchase more continues and try again.
WARNING: The game does seem to freeze a lot on people when they are trying for these achievements in a long endless mode game. Particularly if you get a x10 multiplier on shooting targets. It is suggested that after you get a x9 multiplier, you purposely miss they next target (i.e. do not shoot at it) to prevent the freezing. If your game freezes or crashes to the Start Screen, you will not get any of the achievements (even if you have met the requirements) because they pop after you die in the next screen. You will also lose all of your hard earned continues you used.