On the day he was killed Alexander told Count Mikhail Loris-Melikov that Loris-Melikov's suggested constitutional reform would be discussed in a few days. The biggest part of the suggested reforms was that it would have created two indirectly-elected legislative commissions, although these commissions would serve merely as advisory bodies (The Tsar would still get to enact legislation and could just ignore whatever the commissions said). Now, this could serve as a small step towards constitutional monarchy, but the biggest issue that this requires further, much more substantial reforms from Alexander II. In particular, you need reforms that limit the Tsar's power enough that the next Tsar can't just decide to undo all the reforms with a stroke of the pen. There are two problems with this:
1. Alexander II was not a fan of constitutional/representative government, and didn't believe that Russia was ready for it. Alexander also went through a reactionary turn later in life (which is why the vast majority of his major reforms are during his early reign). This is why the Loris-Melikov reforms were so small, and also why it's hard to see him actually passing much more substantive reforms no matter how long he lives. Which brings me to the next problem:
2. Alexander II probably won't live for much longer. Narodnaya Volya kept coming within inches of killing him, making this one of those "You have to get lucky every time, we only have to get lucky once" situations. Even if Narodnaya Volya fails to kill Alexander he's 62 and the Romanovs had a tendency to die before their 70s. And once Alexander II is dead his extremely reactionary son Alexander III will come to the throne and simply undo anything that promotes representative government or limits the power of the Tsar (and having Alexander III predecease his father doesn't help, because the next in line is Nicholas II).
So in all likelihood Alexander II surviving in 1881 doesn't actually change that much, and we just get OTL's history but Alexander III comes to power at a slightly later date. It's a boring answer, but sometimes history comes to a turning point and fails to turn.