low-budget performance passage-maker
I went through almost the same search a few years ago with a slightly higher budget than yours.
One of the issues in your search will be finding a performance boat within your price range with decent seakeeping ability and at the same time, enough robustness to stand up to the large amount of useage implied by a North Atlantic Circle. To put this in perspective, a heavily used but otherwise normal performance coastal cruiser might sail something like 800 to 1000 miles per year. When people talk about the ''North Atlantic circle'' there seems to be somewhat differing views of that term, but near as I can tell the various routes are minimally 7,500 miles but are generally closer to 10,000-11,000 miles. In other words, something on the order of 10 -11 years of very hard use.
Few coastal cruisers or racers are built to stand up to that kind of use without seriuus modification.
In stock form, boats like the J-35 were aimed at racing with large crews. Their deck gear is on the light side and spread out to provide working space for crew members. Tankage is minimal. Their hullforms are not very tolerant of large heel angles. Thier draft is quite deep.
All of that said, boats like these can be adapted to make reasonable, but very spartan, high performance single-handers. Addint things like tankage and better shorthanded sail handling gear is relativeky easy on boats like these. But adapting any boat to be an offshore capable passagemaker costs quite a bit of money and adapting a full blown race boat from the era of the J-35 can be especially expensive and time consuming. I seriously doubt that a J-35 in structurally sound condition can be purchased for less than $40K. J-35''s in particular are subject to core problems and when you see one for less than $40k the problems are usually reasonably extensive.
Of course the term ''performance'' is very relative. I chose to define it as being a PHRF rating of 102 or below. Most of the candidates that I came up with were selling in the roughly $50K range. A couple boats that I would suggest that you consider for your goals would be boats like the early 1970''s era Tartan 41, or Newport 41 and I would try to find one that someone has already restored and upgraded for offhore use. In the long run that is usually cheaper than doing the restoration and upgrades yourself.
For what it is worth, I ended up buying a Farr 38 (Farr 11.6), which have had an excellent offshore cruising record. Unfortunately they are comparatively hard to find on the US East Coast but are pretty readily available in South Africa within your price range. Of course that adds a lot of miles to your Atlantic circle. Good luck.