how will prior criminal history affect ability to get permanent residency, immigration

Not only does being previously deported make gaining legal permanent residency harder, depending on the reason for deportation, deportation may result in the deported persons being barred for life from re-entering the US even as tourist or other temporary status.

If you are in the US unlawfully after being deported, you may face criminal prosecution if your presence is discovered. Under the statutes, being in unlawful status in the US is generally a civil matter. However, being unlawfully in the US after being deported may be a criminal matter.

If you are outside the US, you likely should remain outside the US until you can get lawful permission to enter the US. Whether you would be able to get lawful permission to enter depends on the reason for your deportation.

If you are already unlawfully in the US, you may have serious problems.

In any case, you likely should review your specific facts with an attorney.