If you really are stuck with 2 layers, then you need to be systematic, otherwise you don't have a chance. Use Manhattan routing, East-West tracks on one layer, North-South on the other, predominantly at least. Then you can route from anywhere to anywhere, using vias to change direction as required, and only moving tracks a little if there's congestion, rather than ripping up and re-routing.
Make the first tracks you put in a grid of grounds, so a track every 10 mm on each layer in the appropriate direction, via'd through at each intersection. This will give you something almost as good as a ground plane. Then place your supply decoupling components close to the IC. Then route high speed tracks next to the ground traces. Then route the rest. Don't succumb to the temptation to put in a ground fill thinking that will solve your grounding problems. All it will do (with 2 layers) is to make it more difficult to see whether all high speed signals have their close ground trace. With 4 layers you do have the space to use a ground plane
However, the cost of a 4 layer board is so close to that of a 2 layer (an assembled, tested, manufactured board, not just the bare board), and the cost of the extra time needed for designing, debugging, and reworking a board constrained to 2 layers is so high (including the cost of getting to market later), that you really ought to push back on this short-sighted requirement.
To be fair, there is one benefit of a 2 layer board, and that is, all tracks are on the surface. This makes rework simpler in some cases where you might want to cut or bridge tracks. However, it's a small benefit to weigh against the other large costs.
A good metric for PCBs is fractional component area - how much of the board area is component footprints, and how much is track. Perform that sum quickly for a few of your company's recent boards. A 2 layer design will require a bigger board, because all of those tracks have to be on the surface. Will the bigger board fit in your enclosure? What's the cost of a bigger board that's cheaper per square cm, compared to a smaller board?
I've lost count of how many times in my long career that the edict has come down from on high that we'll do it this cheap way, and it's ended up costing us far more (especially time to market) because only some of the costs of the different methods were compared.