Alexander did in fact conduct a campaign to the north. His first campaign was against the Thracians and Illyrians, and he marched as far north as the Danube, which his troops crossed. There was a great deal of speculation in antiquity that Alexander intended to embark on a western campaign after his campaign into Arabia, but that his death cut those ambitions short. Most of these speculations are pretty fanciful. Diodorus, for example, says that Perdiccas found memoranda (μνήμη) of Alexander that included, along with quite a lot of building projects, plans to circumnavigate Africa and enter into the Mediterranean by the Pillars of Hercules (an ambition found also in Plutarch), to build a thousand (!) warships larger than triremes (χιλίας μὲν ναῦς μακρὰς μείζους τριήρων ναυπηγήσασθαι), as well as a planned campaign into Libya to subdue the Carthaginians and the barbarians along the coast of Iberia. This is pure fantasy, though. Whether an expedition into Libya was ever seriously planned Diodorus' statement is ridiculous--according to Diodorus the ships that had just circumnavigated Africa were to attack from the Pillars of Hercules while another force attacked from Egypt, building a road from Libya up to the Atlantic! Curtius Rufus also has a similar fantasy, that Alexander planned to march through Carthage into Iberia, whence he would march along the Gallic coast into Italy, march down the length of Italy and sail across the Adriatic to Epirus.
Whatever Alexander's possible future plans for a western campaign were, there was no doubt that his major campaign would be against Persia. A Persian expedition had been brewing for decades. Even before Philip the idea of taking the war to the Persians had been bounced around by intellectuals and orators for some time, and when Philip died he was in the middle of preparing for a Persian campaign. After all, the stated purpose of the League of Corinth was to assemble an expedition into Persia, with first Philip and then Alexander at the head. Besides, what was there to the north and west? The Balkan campaign was conducted swiftly and with few problems--realistically the Thracians and Illyrians presented little threat, and Alexander was only forced to campaign against them, interrupting his preparations for his eastern campaign, because of a revolt.
Just to add on to what's already been said, it's worth noting that Alexander pushed north into central Asia as well. Beyond the river Jaxartes (the modern Syr Darya that flows east of the Aral Sea) a tribe known as the Scythians were proving troublesome. A Macedonian expedition crossed the Jaxartes and inflicted a decisive defeat on the Scythians and pursued the survivors into the nearby desert until supplies fell short. Alexander himself fell ill from drinking fetid water.
It seems that the Scythian campaign was not aimed at conquering lands north of the Jaxartes. Having humbled them in battle, emissaries came from the Scythian king offering peace and friendship. Alexander accepted this offer and while the Scythians continued to bother the Macedonians, those that did so were freebooters out for themselves rather than acting for the Scythian people as a whole. The expedition beyond the Jaxartes had done its job in bringing the Scythians mostly to heel and establishing a clear frontier for Alexander's empire. If Alexander had entertained designs on Scythia beyond merely securing the border, then they were quickly forgotten as he dealt with a tenacious regional insurgency led by a man named Spitamenes. Alexander had more important things to do than push further north into central Asia.
With regard to the west it's already been stated that the last plans found in Diodorus are pure fantasy. Even if there's a shred of truth in any of what Diodorus gives us it's hard to imagine such schemes would have got far off the ground had Alexander lived. However it is worth noting some western Macedonian expansion following Alexander's death. Upon taking over in Egypt Ptolemy sent a general named Ophellas to annex Cyrene (part of modern Libya) and incorporated it into his Egyptian holdings, where it would remain - on and off - until 96 BC. While in command in Cyrene Ophellas even got himself involved in Sicily against Carthage so its undeniable that Macedonian imperialism had something of a western dimension, however small. Of course we don't know if Alexander had any such schemes in mind but if Ptolemy was so quick to organize a Libyan expedition Alexander may have as well.
Follow-up question:
How well did Alexander and the Greeks know about what was up north and to the west?