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There are definitely some folks coming into the Brandeis program with little to no linguistics experience, although I'd say it's more often the other way around -- people coming into the program with lots of linguistics and minimal CS experience. There's also a CS MS program at Brandeis, and a lot of those students cross-register in CL courses, so you certainly won't be lacking in terms of peers in terms of prior course experience. 
Brandeis is not crazy competitive, from what I've heard it's about a 50% acceptance rate. You're right the tuition is high (although not really any different from UW) and they offer very little in terms of financial aid. Going for outside scholarships is probably your best bet. 
The school itself is pretty great. All of the professors are very strong and highly influential in the field, the student body is diverse and motivated, and there's a really nice sense of community that you don't get as much at a program like UW where almost half of the students are doing online learning. The curriculum is as challenging as you make it -- you pay by the semester and not by the credit. It covers a wide range of CL-related topics, in particular computational semantics, machine translation, NER, and applied NLP. There are lots of cool research/internship opportunities in the Boston area, and a lot of folks stick around after graduating. 

Apologies for jumping in really late here, but you may want to consider applying for a master's degree in computational linguistics before applying for a PhD. (Good) PhD programs in CL are very competitive because of the rapidly increasing interest in the field and having a solid field-specific course background as well as having the opportunity to do original research in a professor's lab would really improve your application chances. I'd say to be competitive you need a solid understanding of most deep learning techniques in NLP (MLP, RNN, CNN, LSTM/Bi-LSTM, Autoencoders, etc.)