Before answering your question, let's gather some supporting information first.
Dr. Stephen Strange - Let's discuss about his brief background first, he is a Doctor, a Neurosurgeon, an ordinary man who met with his terrible fate after a car accident which brought his hands into a serious trauma resulting to the loss of the stability of his grip and a difficulty to control them. He was unable to perform any surgeries with precision, therefore began losing his confidence, pride and name. He researched about the people who has the same situation as him but were healed mysteriously, he came up with a person who lost the control of his legs but then is healed and became normal again. Dr. Strange searches to this person and asked how did he become normal again, that is where Dr. Strange was lead up into Nepal to the temple where sorcerers together with their trainer "The Ancient One" train to be powerful enough to prepare for an invasion of Dormammu, a powerful celestial demon who will be summoned by his followers (an ex-sorcerer who have lost their ways) that gained power from the dark dimension.
The Ancient one shows Dr. Strange his inner-self and he can have powers by focusing his mind. After several days of training, Dr. Strange knew that the power he possess is limitless, that he can do much more than he has done. He researched into the library and finds an astonishing discovery, the Eye of Agamotto (the Infinity stone of Time), a necklace that can control time. ** FORWARD ** He found out that the Ancient One also draws power from the dark dimension for the greater good, now that The Ancient One is killed by one of Dormammu's followers, he knew that not all people who uses bad things are for bad things to. After the coming of Dormammu, he uses the Time Stone to rewind everything back the way they were before, but then he realized that Dormammu will still come even though he will rewind everything back, so he stopped the time and trapped Dormammu into an endless loop where Dr. Strange rewinds again and again after being executed by Dormammu, as an agreement, Dr. Strange bargains with Dormammu to set him free from the endless time loop if he will go back from where he came from and never visit Earth again. Dr. Strange succeeded with the trade and saves the planet.
So - Will now Dr. Strange be a part of Avengers?
Answer: Most likely no... well, perhaps not yet.
He did not know that there is a past war between villains and the original Avengers anyway, it so happened that Hulk (Bruce Banner) dropped from space and then delivers the message about Thanos' wrath. He is just an ordinary Doctor seeking for solutions about his hands, if it wasn't for Bruce, he won't know about the past and ongoing war of the Avengers. Since he volunteered to join with Stark to fight for the Earth's freedom then we can call him an Avenger now. But...again, he is not officially an Avenger. We need a confirmation from the Avengers team if he will become a part of it.
(PLOT TWIST)[May Contain Spoilers] However, during the Infinity War movie, he traveled into time to for-see the chances of Thanos being defeated, from over 14 Million Chances, Thanos will be defeated by only 1 chance. So the Dr. Strange's decision giving Thanos the Time Stone is the chance that he has seen to be the only way that could bring Thanos down. But then after Thanos snapped his fingers, half of living creatures in the universe will all be gone, that includes Dr. Strange, we do not know what will happen next but those who were left behind are the original Avengers.
I think that will answer your question now.