Chiggers are are simple thing to get rid of, the bite...maybe not so much.
The itchy bump that is left in the skin is basically a spot where the saliva of the chigger has made a hard tube (basically a straw; thats also why the bump is harder than other bites) and the suck the blood (for simplicity) through it.
Tips I learned while working in the field all this summer:
1. Duct Tape can help A LOT. Tape your pants to your shoes or socks. This can help cut down on both the ticks and chiggers.
2. Spray down with promethrin or deet. Btw, use as directed because it will help more. Word is Vanderbilt just came out with a revolutionary bug spray that will beat out deet and promethrin by leaps and bounds but it will take a few years to get safety certification.
3. Chiggers actually take a while to form the itchy bite (few hours) so one great way of keeping from getting ate up is keep a towel in your truck and rub down with it after you leave the field. Chiggers have extremely weak mouth parts so they will fall off with the slightest brush.
4. After chiggers attach one time, they can't attach again. Therefore once you rub them off they die...thats about the only solice I can offer.
For the bites:
1. It takes time for your body to break down that "straw" that was made. So you are stuck with it for a while. Scratching it till it bleeds can actually lead to an infection.
2. Anti-itch cream can help for the time you have the bites. Bleach SHOULD NOT be poured on the bumps because if gets into the blood it can become a severe health problem