While you and your cat may be able to improvise your own method for going on a bike ride together, there are two products specifically designed for taking pets comfortably and safely on a bicycle ride with you: bike carriers / baskets and bike trailers.
You could just use a standard front mounted bike basket to carry your cat around. Just make sure there's something for your cat to stand on in the bottom (mesh bottom or something you place in it) and you attach a leash and harness to your cat if jumping out and running off is a concern.
On the other hand, there are baskets with additional features for pets. These baskets are generally marketed for small dogs but they'll work for cats just as well.
The baskets above offer some good examples of what may or may not be available in a pet bike basket.
The distinctions between the two options have mostly to do with:
Some baskets come with a wire cage top to keep your cat from jumping out.
You can also find multi-purpose carriers like the one below.
Once you're done using it you can either fold it up and store it or unzip the openings, expand it, and give your cat a place to hang out at home or wherever you are.
Bike trailers for pets are generally made with medium to large dogs in mind, which means they have a lot more room and handle a lot more weight than a bike basket.
While a bike trailer can be a good solution if you have just one cat, it's probably the best solution if you have multiple cats or other pets you want to take on a bike ride (as long they all get along that is). You may want to consider a larger trailer to handle the extra weight and movement if you take multiple pets for a ride.
The best things the trailer above has going for it is that it's inexpensive compared to most bike trailers and it's made by Schwinn (which is a well-established brand that has been associated with bicycles and bicycle components for a very long time).Some features of this trailer include:
If your cat's an indoor cat then you may first want to get your cat leash trained so that your cat is comfortable being outdoors or at least comfortable wearing a harness. The harness can be attached to the basket or trailer to keep your cat from running off.
You should get your cat used to being in the basket or trailer before going out on the road. You don't need the bike attached at first while you do this and you can do this indoors if you want as well.
You may want to get comfortable riding the bike with the basket or trailer attached before taking your cat along. The steering may feel strange with the extra weight of a basket and cat in front. Put a little weight in the basket similar to your cat's weight while you get used to it. Ride around with the trailer without your cat for a while to become more aware of how to navigate the road while pulling it.
What happens if you fall on your bike? You don't want your cat in the basket getting hurt or running off and getting lost. Likewise, the trailer could possibly tip on its side. Think about what to do if that happens before it happens.
Will your cat be startled by noises, bumps, and a lot of surrounding activity? If so, start off somewhere quiet with smooth pavement to let your cat get used to riding in the basket or trailer.
Dogs sometimes like to chase cars or people on bikes. They're probably few and far between these days but be on the lookout and be prepared just in case you can't avoid one.
Reassure and make your cat comfortable. Treats and attention are a good start. Because the basket is in front of you, it'll be easier to keep an eye on your cat and provide reassurance if needed. With the trailer you'll have to remember to look back once in a while (you could get a bike mirror to help you keep an eye on things behind you). You can also place things into the basket or trailer that are comfortable or comforting to your cat.