To modernize the Russian Empire, it is necessary to solve its plethora of weaknesses, which, in our timeline, restricted development. First and foremost, it is crucial to consider that Russian agriculture was, by Western European standards, highly undeveloped. Average yields for the post-Napoleonic period were about 1:4 throughout the Empire (compared to 1:7 in France and Italy and 1:10 in parts of the British Empire). To counter this, the Russians practiced extensive rather than intensive agriculture, and this made up the root of the agrarian crisis, for without increasing efficiency, the growing population would require more than the total of Russia’s arable land. The first step to modernization would be addressing this problem, thereby freeing up people to move to cities (the urban population was a tiny 4-7% of the total). With increased agricultural efficiency and a larger urban population, exports could rise and the various kustaruaia promyshlennost (cottage industries) would grow and take hold.
On the state level, one of the main reasons for slow modernization concerned a perpetual lack of money. With better agriculture and cottage industries, however, the government would be able to fund the development of industry and infrastructure. In our timeline, the construction of the Trans-Siberian railway was an enormous benefit, as it provided some 500,000 tons of grains to be shipped to the European provinces. If this and other railroad/road lines were built earlier, it would contribute greatly to promoting Russian growth. A tariff could perhaps encourage the development of goods within the Empire (there is still debate regarding its effectiveness when used in our timeline), and state sponsorship/foreign loans could be used to establish the first few factories. With capable management and a focus on growth, I believe that the Russian Empire could have been made an industrial powerhouse before the First World War.
It is necessary, however, to remember that significant other problems also existed; namely serfdom, nationalism throughout the various dependent territories such as Poland etc., and opposition and dissent amongst the growing intelligentsia and artistic classes. With some, 23,000,000 people bound to the land, and a fractured, multi-ethnic society, an industrialized Russian Empire is going to have the same (or similar problems) as did Austria-Hungary in our timeline. Pacifying the lower classes will require no or few food shortages (a key motivation for revolution), a strong and ever-present police force, and some unifying factor (perhaps a war, which Russia wins decisively – think of a successful Russo-Japanese War). I hope my thoughts on the subject help!
Please note: Barring a serious revolution, Russia is UNLIKELY to become a constitutional monarchy. Russian Tsars prided themselves upon being autocrats and believed they ruled through divine right. From Catherine the Great to Tsar Nicholas II and including Tsar Alexander II, no Russian Tsar ever would have wanted to surrender their power, as such an action would have been thought of (by them) as unholy and a supreme act of betrayal.