How to make a payment processor like Bitpay? There are some payment processors such as BIPS, BitPagos, BTCMerch, and a few more like them.

In detail, what do I require to get a website like that going?
Can you clarify whether you just need the ability to handle bitcoin-only transactions, or does your service need to be able to trade fiat currencies such as USD or EURO for bitcoins? If so, then you will need to invest in lawyers and find yourself a bank that is willing to extend you a loan, or supply your own funds to get it up and running.

If you're only needing to deal with bitcoins transactions, then I would suggest you learn about how blocks and transactions work at a low level. This is a good site for learning the nitty-gritty of how the blockchain stores data

http://codesuppository.blogspot.com/2014/01/how-to-parse-bitcoin-blockchain.html

Beyond that, understand how to set up a bitcoind server, and send JSON-RPC commands to it using the API explain here

https://en.bitcoin.it/wiki/Original_Bitcoin_client/API_calls_list

Once you know how these commands work, you'll be able to build software that will accept payments. Again, if you need to get fiat currencies involved that's you'll have to deal with legal/regulatory concerns, but it is possible.