In the more recent games, there are NPCs that you can talk to and they will tell you what type of Hidden Power your Pokémon has. According to Bulbapedia, the locations of the NPCs are:
- Platinum: Veilstone Game Corner
- Heart Gold/Soul Silver: Celadon Game Corner
- Black/White: Mistralton City
- XY: Anistar City
- South-Eastern-most house
- Also the location of the Hidden Power TM.
- Omega Ruby/Alpha Sapphire: Fortree City
- Top row of houses, second from the left.
- Also the location of the Hidden Power TM
If you want to find out through combat, here's something I threw together (Disclaimer: there's likely a much more efficient way to figure it out):
1. Battle a Patrat / Lillipup.
Super effective: Fighting. Didn't affect: Ghost. Normal effectiveness: goto 2
2. Battle a Cottonee / Petilil.
Super effective: goto 3. Not very effective: goto 4. Normal effectiveness: goto 5.
3. Battle a Klink.
Super effective: Fire. Not very effective: goto 6. Didn't affect: Poison.
4. Battle a Roggenrola.
Super effective: goto 7. Normal effectiveness: Electric.
5. Battle a Sewaddle.
Super effective: Rock. Normal effectiveness: goto 8.
6. Battle a Woobat.
Super effective: goto 9. Normal effectiveness: goto 10.
7. Battle a Ducklett.
Normal effectiveness: Grass. Not very effective: Water. Didn't affect: Ground.
8. Battle a Sandile.
Normal effectiveness: goto 11. Not very effective: Dark. Didn't affect: Psychic.
9. Battle a Ferroseed.
Normal effectiveness: Ice. Not very effective: goto 12.
10. Battle a Timburr.
Super effective: Flying. Normal effectiveness: goto 13. Not very effective: Bug.
11. Battle a Darumaka.
Normal effectiveness: goto 14. Not very effective: Steel.
12. Battle a Drilbur.
Normal effectiveness: Dark. Not very effective: Rock.
13. Battle a Trubbish.
Normal effectiveness: goto 15. Not very effective: Poison.
14. Battle a Deino.
Super effective: Dragon. Normal effectiveness: Normal. Not very effective: Dark. Didn't affect: Psychic.
15. Battle a Blitzle.
Normal effectiveness: Dragon. Not very effective: Steel.
My thought process (possible types at each stage) is in the revision history of this answer.
The first step would be acquiring a Mienfoo, which can be found in Route 14, Dragonspiral Tower, and Victory Road. The easiest way to breed for a specific nature is to find a Ditto with the nature you want the offspring to possess. From there, give the Ditto an Everstone and put the Mienfoo and the Ditto in the Day Care. The Everstone makes it so that the offspring has a 50% chance of inheriting the female parent's (or Ditto's) nature.
After this, you would just breed over and over until you get an offspring with 31 in either Attack or Speed. Give that baby Mienfoo a Power Bracer if the 31 is in Attack, or a Power Anklet if the 31 is in Speed (this forces that IV to be passed on). Repeat the process until you have a 31 in Attack and Speed, and hope that the ability is Regenerator.
I forget if you need to have a different parent in order to breed the offspring, or if you can just use the same Ditto again.
Checking IVs
The easiest way to check for IVs is to stockpile a lot of Rare Candies. Take note of the Pokémon's stats at each level, and input them into this calculator. The more levels you take note of, the more the calculator will be able to pinpoint your IV. (You can reset your game after testing each Pokémon so you can use the same Rare Candies over and over.)
Best Answer
Besides the one you get from Fennel's assistant in Castelia in Black 2/White 2, you can find Eevee in the Pokémon Dream World. They can be found in the Dream Park, apparently only after you have collected certain amount of Dream Points (I don't have any exact source, but considering its status as a rare Pokémon, I'd say you need 10,000 Dream Points). You have an increased chance of going to the Dream Park when you tuck in a Normal-type Pokémon.
Edit: Source - Bulbapedia