Without trying to put you off, if you've only a budget of a grand, you're going to have to make some serious compromises either in your personal safety, or the quality of the bike. As TallPaul mentioned, the running costs of a 125 really is pennies, but the initial outlay for any new biker, is considerably more than they anticipated/expected and I don't believe your budget is enough. This is not an essay to put you off by any means as bringing more people towards the light of motorcycles is what I'm all about, but I do want to be honest with my views being a biker of 10 years, living as part of a family of bikers.
Because of your height, you'll look like a gorilla on any 125, but if looks are a problem you want to factor in, you may want to check out one of the larger 125 bikes such as a Honda Varadero 125. This will blow your budget and then some before you even talk about a cbt or your gear though. You should be able to pick up a reasonable second hand Yamaha YBR 125, Honda CG 125 or Honda CBF 125 for £800-900 if you do a lot of searching and drive a hard bargain, but they will be few and far between without requiring expense on a new chain and sprocket set, or new tyres and will be a bit of a hunt on ebay, autotrader, mcn classifieds etc. Fortunately, the maintenance costs and parts for 125 bikes cost next to nothing, but expect to stretch to £1000 for the initial buy and first years worth of maintenance as you familiarise yourself with the bike. Bear in mind that there is a very real chance that you will crash this bike like the vast majority of us will have, whether it be from a simple miss-footing whilst rolling to a stop, a low speed accident with a few bruises or broken bones, to a more serious life threatening incident. With any luck, you only have to experience the embarrassment and damaged pride from dropping it, but you ought to be prepared to spend money to fix both it and you as well.
As for your riding gear, you should be expecting to pay another £300-500 at least, with your helmet being one of the most expensive items you buy. You should not cheap out on a helmet as this is one of the most important safety items and is the only thing that is going to save you from brain damage or death should you have a nasty accident. I would only ever recommend buying named brands as these tend to have the minimal compromises as far as rider safety is concerned. With most things, the more you spend, the more you will get out of the helmet. It is a consumable though, so any drops and you should look to replace it, and you should also look at replacing them every 5 years or so due to UV damage from the sun. Each manufacturer have their own recommendations based on the materials used. You can expect to pay anywhere from £130 from HJC, to £350-£500 for a non-graphic Arai, Shoei or Schuberth. With helmets, cost isn't quite everything and you should absolutely choose a better fitting helmet than the more expensive one and should only buy if you can try the helmet first, having worn it for 5-10 minutes to ensure that there are no pressure induced headaches or other issues with the helmet or your vision. You should also have it tested for fitting by a trained shop assistant as well to ensure that there is not excessive roll/free play with it on your head until such a time that you know exactly what you're doing when it comes to shopping for bike gear. As far as jackets, gloves and trousers go, try and pick out kit that has some form of shelltex, goretex or other similar waterproof lining built into the garments as waterproof over-kit tends to cause more sweating in my experience. The problem with this is as the quality and durability of these linings and the garments increase, so does the price considerably. As far as boots go, there's no need to go full motor-x armoured, but you should at least be looking at some form of shin and ankle armour which will come built into any reputable motorcycle footwear. Sidi are pretty much the go-to for most, but do try out the other brands as well, ensuring you have enough movement to pitch your foot up and down to operate the gear lever and rear brake, but not too much that in the event of an accident you are less guarded for doing serious damage to your foot. All boots will break in, so bear in mind that both stomping around like an elephant and operating the motorcycle will get lesser and easier over time.
