What cable:
Recommended choices are Webro WF100 / Triax TX100 / Labgear PF100. These are all good quality well-shielded coax cables made from copper rather than inferior steel/aluminium combinations. Expect to pay around 50p per metre, so hardly breaking the bank. Webro WF100 is also available in white if it helps with blending in to room aesthetics.
All the above have a physical diameter of around 6.5 to 6.7mm.
Webro also makes a thinner version in a twin (shotgun) configuration made popular by Sky satellite installations. This is WF65. Each leg of the twin has a diameter of 4.6mm. The twin cable can be split in to two individual conductors.
Yes, you'll end up with a spare length of the stuff. No, unless you can find a retailer willing to split before you buy then there's no way around this. Even if you do find someone, the cost of the labour will probably be more than you would save in material costs. Coax isn't expensive. It's cheaper to get the right stuff from the word go rather than buy rubbish, then waste ages trying to fix the problems it causes, only to realise you then have to buy the good stuff anyway and re-do the job.
WF65 is also available in white.
IMPORTANT NOTE: Webro makes each of the above in two versions. The one to go for is the all copper version. That means that the shielding is a copper foil tape + copper braid sheath. The one to avoid is the Aluminium/PTFE tape + aluminium braid. I mention this because if you're buying from Ebay there are lots of sellers who conveniently 'forget' to mention which version they're selling, so you think you're buying the good copper stuff but get supplied the inferior aluminium grade.
Stuff to avoid:
TV Cable extension kits. Invariably they're crap. The cable might be thin but it's also the worst quality. You're just inviting problems with signal loss and interference. It just ain't worth it. Run away. Run away.
Anything sold as RG6 or Low Loss or Sky/Virgin/Freeview HD/4K where the cable has aluminium (silver-looking) shielding. The vast majority is a cheap grade 6.5mm coax using a copper coated steel (CCS) core conductor and aluminium braid/plastic/PTFE tape combo shielding. Cheap aerial jobs are done with it because (a) it's marginally cheaper and (b) it'll fail outdoors far quicker than all copper coax so there a chance of some repeat buisiness for the installer. It's bad bad stuff. Avoid.
Can you do it?
From a technical point of view, yes. It's not difficult for the average semi-competent DIYer. The chances are that the signal will already be pretty stong in a block of flats as there'll be a professionally installed aerial distribution system to ensure a decent signal level to each flat. For that reason you won't need a powered booster unless you buy the cheap TV extension kit or crappy RG6 I've just told you to avoid in the paragraphs above.
By spending pennies more to get decent cable quality you won't need to spend £15-£20 on an amplified splitter.
See how that works?: Spend enough here, and save big there. That's a good thing.
What you'll need is a passive splitter.
SEE LINK. This will give you two identical signals without reducing the quality (Q) of the signal. There'll be enough strength (S) to cope with splitting.
You'll notice that the splitter has sockets for the screw-on type connectors you might have seen on satellite gear. This is pretty common now. All you need is some of the F-plugs (satellite connectors) instead of using TV coax plugs. F-connectors are easier to fit correctly and give a more secure connection. If you buy WF100/TX100/PF100 cable, then get RG6-sized F connectors. If it's WF65, then get WF65-sized connectors. The screw-threaded sockets on the splitter are a standard size regardless of the plug size, so any plug size will fit. The two different sizes of plug are for the different diameters of cable.
Can you do it from a tenancy point of view? Only your landlord can answer that question. Some tenancy agreements don't even allow Blu-tack or drawing pins to be used on the walls. Where the flat is purchased rather than rented then this shouldn't be an issue.
As far as the act of splitting goes, you're doing nothing different than would be the case looping the aerial signal through a Freeview recorder and on to the TV. It shouldn't affect other tenants.