Tenant problems in Nassau County NY

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Hey gang, I have a duplex in Nassau, I live downstairs and rent upstairs, it's been pretty good until a new couple came in, they been here 2 mths and already missed payment on the 2nd month after nicely talking to guy I found out he had a problem with his employer and his check bounced twice and was thinking about pursuing legal action toward his employer (seems like he has no residency papers, illegal) so gave him the benefit of doubt, 3 weeks later he has given me 3\4 of rent and said he will give me the full amount and the full rent come April, I figure ok, we all have that bad luck so I said ok besides he has given me 3/4 rent so I figured he is trying. the apt has always been a mess, every time I go upstairs I tell them to keep the place clean and yesterday I went up for a bath problem and it looks worst than b4 in fact I saw a cockroach upstairs and told him to best keep things at peace and would like for him and fam to move out asap, because of other small related issues, he said he needed a couple of mths but I said l would give him until next mth but the sooner the better, even offer to give him his dep or part of it back (depending how apt looks when they leave) but I heard some horror stories in Nassau about evictions so can anyone fill me in on Long Island evictions or what best way to pursue this would be appreciated, I hear some landlords can be A-holes and I just don't see why you would b like that especially in NY. 

Also I seen this 45 days eviction attorney sign in the Queens area I wonder if anyone has heard about this? tnx.