Hardware store stuff ....
Get TSP (trisodiumphosphate) from a hardware or paint store. Dissolve it in HOT water until no more crystals dissolve and then slosh on, let sit, then with a soft brush scrub away the grey, the black specks and mildew. Flush with lots of water. If the teak has been neglected for a long time you may need to repeat with the TSP before bleaching with oxalic. Follow the safety precautions on the TSP and oxalic packages. From the same hardware or paint store get oxalic acid (wood bleach) ... do the same as above but brush on. The bleaching will be almost instantaneous. Throroughly flush with several flushes of hot water. Consider a 'resinated' teak oil that contains UV inhibiters. A resinated teak oil contains approx 25% varnish but is very easily stripped in the future. "NuTeak" by Marykate is a good one. Apply it THICK (DONT RUB IT OFF apply it like you would apply varnish .... and with several coats if you prefer) .... will last an entire season or more. When it needs stripping (most oil finishes will turn dark eventually), simply soak with TSP, etc. as above. If you cant find a resinated teak oil, simply add 25% oil based varnish to your favorite oil. If you want to get 'carried away' ... apply 6 or seven coats, let fully cure in the sun for several weeks, then flatsand with 600 wet and dry paper then handrub with rottenstone and water to result in a gloss that will EXCEED a high quality varnish.If you want a more 'permanent' finish similar to oil consider "Teak Wonder or Semco Natural". I like to mix the two 50:50 .... or for teak decks/seats,etc.: 1/3 Teak Wonder, 1/3 Semco, 1/3 Olympic Deck stain (carmel). hope this helps.