I was able to get it to work using just NSUrlConnection. You must first base64encode
NSString *keyString = [NSString stringWithFormat:@"%@:%@", BING_SEARCH_API_KEY, BING_SEARCH_API_KEY];
NSData *plainTextData = [keyString dataUsingEncoding:NSUTF8StringEncoding];
NSString *base64String = [plainTextData base64EncodedString];
Setup your request
NSMutableURLRequest *req = [[NSMutableURLRequest alloc] init];
[req setURL:[NSURL URLWithString:searchUrl]];
NSString *authValue = [NSString stringWithFormat:@"Basic %@", base64String];
[req setValue:authValue forHTTPHeaderField:@"Authorization"];
Make the request
NSError *error = [[NSError alloc] init];
NSHTTPURLResponse *response = nil;
NSData *data = [NSURLConnection sendSynchronousRequest:req returningResponse:&response error:&error];
Look at the documentation about how to form searchUrl, and then process data according to the format you specified in $format= (I used json, so mine looks like):
NSDictionary *json = [NSJSONSerialization JSONObjectWithData:data options:NSJSONReadingMutableContainers error:&error];
I left out error handling, dont forget to add that by checking response as well as error.