I have a British Passport and would like to visit my sister in the USA, but would like to stay for longer than 90 days. How would I go about this?
You can normally stay for longer than 90 days if you have a B visa. See, for example, https://travel.state.gov/content/travel/en/us-visas/tourism-visit/visa-waiver-program.html:
May I apply for a visa instead of using the VWP?
Yes, you may apply for a visitor (B) visa, if you prefer to do so or if you are not elegible for VWP travel. ... Also, if you intend to stay longer than 90 days, then you need to apply for a visa.
With a B visa, you will normally be admitted for an initial period of six months, which can be extended. If you stay for longer than 183 days, however, you may become a tax resident for US income tax purposes, which you probably want to avoid, so you should consider the implications carefully before deciding to stay that long.
The duration of admission is governed by regulations found at 8 CFR 214.2(b)(1) and 8 CFR 214.2(b)(2):
(b) Visitors -
(1) General. Any B-1 visitor for business or B-2 visitor for pleasure may be admitted for not more than one year and may be granted extensions of temporary stay in increments of not more than six months each, except that alien members of a religious denomination coming temporarily and solely to do missionary work in behalf of a religious denomination may be granted extensions of not more than one year each, provided that such work does not involve the selling of articles or the solicitation or acceptance of donations. Those B-1 and B-2 visitors admitted pursuant to the waiver provided at § 212.1(e) of this chapter may be admitted to and stay on Guam for period not to exceed fifteen days and are not eligible for extensions of stay.
(2) Minimum six month admissions. Any B-2 visitor who is found otherwise admissible and is issued a Form I-94 (see § 1.4), will be admitted for a minimum period of six months, regardless of whether less time is requested, provided, that any required passport is valid as specified in section 212(a)(26) of the Act. Exceptions to the minimum six month admission may be made only in individual cases upon the specific approval of the district director for good cause.