How can I go from Canary Islands to the Caribbean by sea in October?

I don't have much good news for you. According to Freighter Trips (and various other sites I could find):

Many people who are taking a freighter voyage as part of a long term stay abroad, or a permanent relocation, ask about the possibility of taking their pet with them.

For animal lovers, this would certainly be a dream solution to their relocation problems, but sadly I believe it is now impossible. Freighter shipping lines will not allow pets to stay in a passengers cabin, or anywhere else onboard ship.

The only transatlantic cruise ship that seems to accept dogs is the QM2

The Queen Mary 2 operated by Cunard Lines is an approved carrier in connection with the Pet Travel Scheme (PETS) on Transatlantic Crossings. Passage in the kennels is subject to size restrictions, availability and the requirements of the Pet Scheme. Pets are accepted on all Transatlantic crossings, except the first and last crossings of the year.

However, that goes from Southampton to New York. You'd have to get to the UK and then drive down the US and cross from perhaps from Key West or somewhere else in Florida.

For air travel, you can do Tenerife to St. Lucia or Barbados via one stop in London Gatwick which doesn't sound too bad if you get appropriate advice from your vet.

There is at least one cruise ship that travels from The Canaries to the Carribean but like all the other cruise ships except QM2, they only allow service dogs.

I don't know of any regular scheduled shipping from the Canaries to the Caribbean. However I have a couple of suggestions for things to try out:

1. Repositioning cruises. Sometimes cruise lines have to move a liner from the Mediterranean, and they offer travel on that cruise leg at a fairly cheap price. Some might stop at the Canaries. These only happen at specific times of year, so this won't help if it doesn't fit your schedule. These are often hard to find, so you will need to check each cruise line.
2. Hitch a ride on a private boat. Boat owners, especially sailboat owners, are often looking for people to help out handling the boat, and offer free passage in return. The passage from southern Europe to the Caribbean is a popular one, and a stop at the Canaries is common. To find out of people are doing this, go down to your local yacht marina and ask around. However you won't be able to take lots of luggage, and pets may be an issue. On the upside its very cheap.

As your question is posed, it's almost impossible to answer: there are dozens of island nations in the Caribbean, along with another dozen countries that border on the Caribbean Sea. It would be better to decide first on your destination, one which would permit you to move/emigrate, and allows the import of pets (with or without quarantine). Global pet relocation is so common now, that there is an International Pet and Animal Transportation Association, through which you could explore transportation alternatives and find the shipping options available from your location to your new home.