How does one scale encryption strength upwards from 256-bit? I have seen many examples of encryption up to about 256-bit. But how does one programmatically scale the logic upwards in a language such as PHP or Java to say 1024-bit or even 4096-bit and higher?

Are there any websites that explain the workings of a very large encryption on a modern 64-bit computer? For example what kind of numbers (Long Int, Double, etc.) would one use and how can one even extend that range and still keep it computable to the computer?

Would this scaling's security be compromised by the limitations of the processing power available, i.e. 64-bit processor?

I understand that time is a factor in very large encryptions, but in this case time doesn't matter that much.
As @David Schwartz said, using something like 3AES should work to scale up the security (but see my comment on the meet-in-the-middle attack). But seriously, why would you even need 512-bits of security in a symmetric algorithm? According to the analysis on wikipedia it would take 50,955,671,114,250,100,000,000,000,000,000,000,000,000,000,000,000,000 years to bruteforce 256 bit AES (given a few assumptions on speed). I doubt you have any data that needs to be secured for that long.

As far as whether the CPU limits the security strength, it doesn't. There are libraries out there which provide "infinite" precision numbers (i.e., gmplib for c/c++, BigInteger and BigDecimal for Java, etc). For more on computing with large numbers, see chapter 14 of the Handbook of Applied Cryptography (HAC). Internally, these libraries use things like long long int or even plain int stored in an array. HAC has the algorithms.