We are on our 2nd Basenji, he'll be 3 in February. We got him when he was 5 months old. He was bitey and chewed everything not his own.
Try to discourage this behavior immediately because if it gets out of hand it will continue for a long time and eventually he will seek other family members to play bite with that he knows he can get away with (smaller children, etc ).
Get him a lot of toys, make sure to correct his behavior at every opportunity. Do not ever rough house play with the dog using your hands. Get a basket and keep his toys in the basket so his toys are in his pile, not scattered around the house. He will chew the basket too, get another one. Keep his toys together.
I disagree with using crate if that's where his bed is when you are not home and leave him in.
If he bites you, you can hold him by sides of neck scruff and tell him sternly "NO"! An alternative is to lay him on ground and hold him down and say "NO"! You'd be surprised but at 4-5 months he can already be trying to be alpha male over you, so one exercise you can also consider for this is to hold him in front of you and keep a long stare until he looks away, your family members must be the alpha and your dog needs to start learning the pecking order. Some may not agree with any of this, but I'm not here to judge. Basenji's can he difficult breed, some can turn out very very stubborn and this is a very determined breed. Also ask your breeder for recommendations.
The suggestion of socializing with other dogs is good idea so he learns more how to behave around others.
Good luck, it will get better over time.
I adopted a Basenji mix at 8months old with this issue. Try and remember that they are like a child and can't always make the connection between what they are doing wrong and what you want them to do.
From a psychology stand point for my dog she was clearly raised away from her mother and litter-mates so the rough play thing was a huge issue. I did two things to work on it and I can't guarantee it'll work for you, but reading this might help your perspective on how you may need approach training a behavior. 1st) I worked on other things like the command "leave it" (when she would take a sock, i'd say a firm "leave it", and then take it and put it where she couldn't get it) I'd also add in a verbal "app!" right before "leave it" (the reason for this is i can apply "app" to any new or old command to get her attention that she is doing something wrong.). The idea is that when you pup hears "app" they should stop what their doing and pay attention, be it your arm if she is biting you. From there you make a new command "app" "no bite".. after awhile "no bite" become the command to not bite... To approach the squealing and the sympathy reaction exposing her to other dogs when age appropriate might help learn those social ques. This was one of the hardest thing my 8month old basenji mix had to learn having NEVER being exposed to another dog prior to her 1 week stay at a kill shelter before adoption when she was already 8 months old.
LASTLY and probably the most important thing is, IT TAKES TIME, and you're doing great! You didn't learn to read, walk, use a phone, or ride a bike in one day or afternoon. It takes days, weeks, and sometimes months of routine for it to stick, and even after they "get it" you have to continue to reinforce it. You'll make mistakes and sometimes a training method doesn't work so you have to change it up.
I hope this helps! Feel free to ask this forum anything, its a great bunch of people with A LOT of combined knowledge.
Some good advice already here, and remember that you have to figure out what works for YOUR dog and YOURSELF - part of the magic of having a dog I guess (or kids). Our first pup, when in puberty could bite hard, too. We found that sound or verbal commands did not work (for dog and ourselves). We took to body language, standing firm (Yang in Chinese terms, putting some energy pressure, but not too much), silently, turned towards the dog, and turning with her movements, so she knew we were actively involved in the scene. As soon as she got it and stopped, we would go soft (Yin) again and turn away to take away the pressure. It took patience and practice, but we learned a lot about how these non-verbal interactions work and what makes our miss Lela tick. We still prefer - whenever possible - body language over verbal commands.
If you watch what dogs do with each other, you will notice that although siblings (peer group) might yelp when the biting escalates, Mom generally won't, but will growl and frequently she will pin the pup down and sometimes snarl in his face if he's too aggressive. Mom isn't peer group, she is boss. I prefer to be boss, rather than peer group. 
I don't like time outs much because the pup may not accurately interpret the action. If you use these, make sure to indicate what is coming immediately when the bad behaviour occurs (a word, a growl, whatever), otherwise the negative punishment is too far removed from what caused it. And yes, it can cause the pup to dislike his crate or leash if either are used to accomplish the time out.
Is it possible that he is teething ? at 4, rising 5 months, I'd have a good look at his gums and make sure there is no unusual redness.
I've never used time-outs or any kind of 'after the event' discipline. Dogs (and particularly Basenjis) have no conception of cause and effect. Instant reprimand, a short, sharp 'NO' and then ignore him. No more playing. End of game.
Never let him think he has your undivided attention after he has bitten. Avoid rough games and situations where he thinks biting is acceptable. Some of it may be attention seeking and your squeaks reinforce his idea that it is FUN and you agree with him.
He won't like being ignored and a combination of a display of your displeasure - NO ! - followed by a total loss of your attention should, over time, bring it home to him.
But check his gums too.
My basenji is not typical but I will tell you something interesting I have discovered with my basenji. Once a week or so she seems to get agitated and starts biting my arm. I take her to the backyard and run with her for 5-10 minutes. She does the Basenji 500 and runs and runs and runs. Then she goes poo. After that she is fine. I am guessing that she is feeling something but doesn't recognize what it is. After some running she is ready to go.
This may not be the case with your basenji but no harm in trying right?
Hey everyone! Well thanks so very much for taking the time to respond to us. My wife and I have been working hard with our little mister all week and it's paying off. The point made about agitation due to having to go potty was a good one - sometimes we realized that he might have to go and was anxious because of it so we took him out. Several times that was definitely the case.
Other times, he's just excited (usually in the morning first thing and after dinner with huge naps in between) and we try and work the energy out. We live in a condo building but we've started to take him to run full-speed in the halls between the two of us and he seems to really enjoy it! We can't believe how fast our little guy is!!
We encourage playtime and gentle little mourning/bites are okay, but when it's too much he gets "ENOUGH" and right to time-out. He's really stopped biting altogether so I think it's working!
Plus the additional interest/reward of training him new tricks keeps him occupied and gives him lots of positive feedback.
It's a long road ahead but so far we're starting to see progress so thanks to everyone for your input!!
I know it's kind of late to chime in here, but, I feel I need to share my experience with you. We got our first basenji in March '96 and our 2nd in December of that same year, brother and sister from year apart litters. They could not have been more different. Our female was an angel, an occasional basenji high jinx, but her brother, imp all the way. He was a biter from the get go. Like you, we tried everything. What finally worked or us, immediately stop play, stand up, fold arms, out of reach, and turn our backs from the puppy. He quickly got the message, that biting of any kind would not be tolerated. I really need to caution you here, any kind of biting, even in play, should not be allowed, as this most likely will become an issue for you in the future, especially if you plan on having children in your home. He will think that it's okay to put his teeth on/into your skin. I ought to know as our Bolt, although very much loved in his original home, after 6 years, had to be rehomed due to biting issues. We are his 4th home. Not everyone can cope with an adult biting dog. It's best to deal with it now.
They are very smart dogs. I say they are more like a 2 year old in a dog suit. BUT, what works for one does not mean it will work for another, and sometimes, it will only works once. They can be very challenging, but the reward, that, that is limitless, and why so many of us have them. If you ask most of our relatives, they simply think we are nuts for having 'crazy' dogs!
Hey all!
So update for everyone. There is light at the end of the tunnel! Our little Bailey has stopped being so bitey and directs most of his attention to his toys. We let him run like a Wildman in the the hallways of our condo (it's freezing outside right now!) and that seems to tire him out. We still tolerate little nibbles because it's playtime but anything that's too hard get stopped right away. He knows it and comes over to lick instead with his guilty-pooch eyes 
Thanks again for all the advice! Next step, figuring out why the heck he's started waking up at 4am and whimpering when he was totally fine and sleeping all night before!
Cheers,
Take him out last thing at night so he can make himself comfy ?
I have to take mine out - wimps - they won't go if I don't accompany them. Yes, its freezing and yes its dark. But Keeper's close-up vision is going going gone and he hates excursions into the night without Mom and although I'd much rather stay indoors, if I want to avoid being awoken, middle of the night there is no option.
Frankly its a small price to pay for the comfort of two old Basenjis.
