The weather here will, for the most part, allow you to walk in all four seasons, but the weather can be dramatic enough that you will not always want to. Snow, freezing temperatures, wind, and rain are all a part of New York winters. In the summer, the biggest enemy is the heat and humidity.
Bedford Park is not, by any means, a neighborhood that you would describe as hilly, but it does have some hills in it. The hills are mainly gentle inclines that are perfect for adding a bit of extra work into your daily walks. You can easily avoid the hills if you have no interest in getting any serious exercise as there are plenty of flat areas in Bedford that make walking a pleasure.
Bedford is definitely an urban area, and as such, you can expect to do most of your walking on concrete with little in the way of grass surfaces. There are trees in some areas, including some that are tall and provide a bit of shade. If you are walking on warm days, you might consider planning out a walk where most of the sidewalk is protected from the sun. It will make it much more comfortable for your pup!
You also have access to Harris Park and Jerome Park to the west of the neighborhood. Jerome park includes a 1.5-mile jogging path which goes around the Jerome Park Reservoir.