22 Tamuz 5775
There was no Nefesh b'Nefesh (NBN) when I made aliyah. I just contacted the Jewish Agency in Denver where I lived at the time. The Jewish Agency is still in the aliyah business and in fact, everyone who uses NBN's services must still go through the Jewish Agency.
From everything I've heard from those who have used NBN services, they are very discouraging of those most willing to give up on life in the Diaspora - the poorer segments of Jewish society - and even those whom they deem to be a good risk of their funds don't get the kind of support they were expecting after the fact.
Their website, however, offers a wealth of information that is free for the taking - like this page which explains what you need and where you need to go in order to "make aliyah" from within Israel.
If you can afford the (one-way) plane tickets over and a temporary place to live til you get processed, you don't need anyone's approval before coming to Israel. You just come and then apply for immigrant status at the Ministry of Interior.
However, be aware that the people you are dealing with at NBN present aliyah like it's only for rich people - pilot trips, so much money in the bank, arrangements to bring over your car, etc, etc, you get the idea. It reminds me of those studies that claim it takes 1 million dollars to raise a child to maturity including the cost of his college education. Poor people all over the world have babies with no money and poor Jews can make aliyah without money, too. It's hard, yes, but it can be done. How badly do you want it???
I recommend you read this two-year-old Haaretz article on NBN before deciding whether to approach NBN or not. Just know that it's not necessary. You can go directly to the Jewish Agency and by-pass NBN.
(See this nice Jewish Agency for Israel (JAFI) online brochure for a description of the process, rundown on benefits offered to ease absorption, and important contacts.)
It says: "We recommend that you schedule an optional meeting
with a Nefesh B’Nefesh representative to explore their
programs and services and to participate in their PreAliyah
sessions and seminars." But, it's not required.
If I had to give the aliyah shaliach at the Jewish Agency a reason for why I did not want to approach NBN, I would be very honest and tell them that I had it on good authority that they are very discouraging of poorer Jewish families making aliyah. I would tell them how determined my family is to go and have a successful klita and that you believe that having endured hard times in America has already prepared you for living with less in Eretz Yisrael. If the Ethiopians can do it, the Americans can, too.
Read this brochure carefully to understand your immigration rights as a Jew and don't let anyone mislead you, intentionally or otherwise. For example, if you are ready to leave for Israel and someone wants you to wait for a NBN flight, it's not necessary: "Once you have received approval you are ready to finalize
your Israel program plans and book your flight. Your
“Aliyah” flight is paid for by the Jewish Agency for Israel
and facilitated through Nefesh B’Nefesh’s group and
charter flights. Individual flights are also covered."
To me that says, normally you would be lumped into one of the NBN flights which are only scheduled a few times each year, but if you choose to fly separately, that is also your right. (That's how my family came before NBN.)
Let me know in the comments if there is anything else I can look up or a general question I can get answered for you. I'll do my best to help, but specifics will more than likely have to be answered by your local aliyah shaliach. People are welcome to share their own experiences, what they would do differently if at all, etc.