1: Hacking the brain requires using the brain
The best way to make a believable "brain hack" is to treat it kind of like a computer hack. You can not force the brain (or a computer) to do something that it is not designed to do, but you can hijack the systems that are there to achieve some unintended functionality.
Research shows that the main purpose of dreaming seems to be focused on learning through simulation, making it an ideal brain system to hijack.
According to the field of Behavioral Psychology, thinking is indistinguishable from action when it comes to learning. So, anything you think can be learned, and anything you learn must be thought. That means that if the drug can make you think about and feel all of the memories you want to implant, your memory treats that the same as living through those experiences.
The reason this presents a problem for your drug is that IF you take it while awake, you will start to hallucinate the memories that you are implanting while you try to live your life. This will make focusing on the real world exceptionally difficult during the learning process (Caution: do not operate a motor vehicle while taking this drug). It will also mean that the real world could corrupt your new memories. If you try to do your taxes while experiencing a memory of skydiving, then you may well remember doing your taxes while skydiving. In this way, dream learning lets your brain absorb complex memories without outside interference.
2: Time becomes less of an issue
Dreams happen at the speed of thought; so, they often cover more simulated time than what has actually transpired in real life. People often awaken from dreams that only lasted 15-30 minutes and report having experienced hours or even days in thier dream world. Unhindered by reality, this accelerated "dream time" is probably the limit of how fast the human brain can absorb new experiences.
Part of this is because thought is not limited by the real world so you can actually process the "plot" of a dream faster than you could live it, but part of it is because imagination/memory works by creating memes woven together by logic instead of like a digital video or audio clip where the whole of the experience is recorded. So, part of that acceleration comes from a lack of complete details. The more detailed your dream learning, the longer it will take.
All this said, even though you can live out experiences in accelerated time, a longer memory or set of memories could still take hours of time to record, and it will take a lot of repetition to get most of it since we normally remember less than 10% of what we actually experience.
So, doing it while you sleep may be necessary because you will need relatively long periods of undisturbed brain time to "upload" a memory in any sort of detail.
Some important things to remember about injected memories
Imagine for a second that the following memory is injected into your head.
You stopped for ice-cream in the park and a friendly looking dog stole the ice-cream which made your kid cry. You tried to comfort the kid, but you could not get them to stop crying.
For fun, you might want to write down how you picture it in as much detail as you can before reading the rest of this answer, as this will help illustrate my point later.
The thing about memory not being a video is going to be very important when realizing how different people will absorb implanted memories. No memory is a self-contained event in your brain. Studies into false memories show that everything we remember is interwoven with all of our own personal knowledge and experiences, and that they can actually change each time we replay them in our heads because each replay is actually a new experience. So, if you inject an identical memory into 2 different people's heads, and later ask them to recall it, they will come up with 2 very different narratives of events based on thier own other experiences.
For example, lets say that both Bob and Mary receive the same implanted memory I described above and were asked to recall the memory later, the stories they may tell could sound like this:
BOB: I went to the park with my 2 kids. As we walked along the path near the big white gazebo, we passed under the giant oak tree where the ice-cream guy usually parks his cart. I bought a cookies and cream ice-cream cone to share with the the kids. As we were walking along, a big brown Labrador ran up and snatched the ice-cream right out of my hand! My younger daughter cried so much, I had to carry her the rest of the way home.
MARY: I brought my nephew to the park. There was guy selling ice-cream. I decided to get some for my nephew. As we were walking along, a little beagle came up and stole the ice cream from little tommy's hand. I offered to buy him another one, but he just stood there crying for a long time until eventually I walked him home.
Why are these two memories so different? Because humans make shit up when we remember things. Whenever a memory is incomplete, you inject a bunch of personal biases, assumption, and other memories to create a semi-reliable picture of the event. Even though most of what we remember is made up, it is all based on reasonable assumptions by combining what we know with what we remember to create a usefully completed narrative that is probably true.
Bob and Mary Explained:
Because they remember a kid crying, they both understand that they were there with 1 or more kids, both people impress the most likely candidates into that role. For Bob, it is his 2 kids, because if he went to the park, he knows he'd probably bring both, but for Mary, it was her nephew. She does not have kids, but sometimes she takes him places; so, that makes the most since to her.
Next, is the environment. Bob knows about an ice-cream guy who works in the park; so, he can picture not just that guy, but the whole environment around where he'd expect to run into that guy. In contrast, Mary does not know where she would get ice-cream from in the park; so, she works under the vague assumption that somewhere in this completely un-detailed setting, someone sold her some ice-cream.
Next, each person makes assumptions based on thier own habits. Bob knows that he would normally buy a cookies and cream ice-cream cone to share with the kids. Since he does not remember anything specific about the ice-cream, he assumes this as the default. In contrast, Mary does not normally eat ice-cream; so, she assumes she just bought something for the kid she was with.
Next, assumptions start to compound more as what we've already pictured starts to take hold over what we picture next. For Bob, he remembers sharing the ice-cream, meaning that he must have been holding it, which means the ice-cream had to have been stollen by a big dog to have been able to reach his hand. So, when he pictures a big friendly dog, he imagines a Labrador. But for Mary, she pictures the ice-cream in the child's hand, so if a big dog like a Labrador approached little Tommy, 'friendly' would not be the first thought to pass through her head. So, she instead pictures a smaller friendly dog and comes up with a beagle instead.
Lastly, how each person would comfort the child is different. For starters, Bob has two kids in mind; so, he default picks the one that best fits the story. Bob knows that when his younger daughter cries, his first reaction is to hold her, and if he can't calm her down, he knows he'd just carry her home. But Mary pictures calming down a child as a problem solving exorcise; so, she imagines herself talking to little tommy and trying to offer solutions to the problem.
Because our brains do not record events in full detail, and can not recall events without projecting our own biases onto them, it will be important in your setting to remember that each person's injected memory should be a unique experience, even if it is based off of the same source material.