That's relatively easy. Check with the local city or county tax department to determine who owns the property. Usually, Googling "_______ tax assessor" will get you there. Most jurisdictions have the property information online, including the name and address of the owner of record. That'll give you the name and address of the owner.
If that doesn't work, again check with the local courts--whichever one was involved in the eviction. The owner filed an unlawful detainer action (that's the official term for eviction) against the tenants, and that'll be public record. Again, that'll give you the name and address of the owner.
If that doesn't work, knock on the doors of the neighbors. Explain that you're trying to find the owner of the house. And explain why. The neighbors will be delighted that someone would like to move into the vacant house.
So, one way or another you've tracked down the owner. (There are other ways, too, but the suggestions above will work 99.9% of the time.)
Then you approach the owner. Call or write a letter. Explain, just as you did here, that you'd like to purchase the house. Then negotiate.
You would be much better off using a Realtor for the process. However, investors go through the same process all the time, usually without a Realtor. Still, you need to know what the house is worth, and the other information that will help you make an offer.
And you might be able to get a good value. It's likely that the owner is now seriously motivated to sell. Owners don't like the eviction process; they also don't like empty houses. Remember: You're offering a solution to the owner's problems. That puts you into a very strong negotiating position.