It will be rare that an expedited petition is appropriate - the case law concerns instances where only compulsory liquidation is possible for a type of entity but the usual process would cause significant prejudice to creditors, e.g. credit unions. The general procedure for winding up petitions must be followed and pay particular attention to the contents of the petition (r7.5) and any regulator consent that might be required (PRA and FSCS for a credit union for example). An application for an expedited hearing is made pursuant to the Court's inherent jurisdiction and also the case of Re Marches Credit Union Limited  EWHC 1731 (Ch).
As well as an expedited hearing, a direction should be sought pursuant to r7.10 that the petitioner have permission to not give notice (this rule starts, "Unless the Court otherwise directs…"). The certificate of urgency at para 9.1 of the Insolvency PD will need to be completed at the time the petition is filed.
If an expedited petition is considered because of a perceived risk of dissipation of assets or risk to the public of continued trading, then provisional liquidation may be more appropriate.
An application is made pursuant to s135 of the Act and the general rules on applications will apply (r1.35 and 12.6-12.13). Also it will be necessary to have regard to r7.33 - 7.39 which sets out who is entitled to make an application, what the evidence in support of such an application should contain, the requirement for an IP's consent to act, and requirement to serve the OR. The applicant will have to deposit such sum as the Court directs to cover remuneration and expenses and do so within 2 business days after service of the order.
Applications are made pursuant to s127 or s284 of the Act. General rules on applications (r1.35 and 12.6-12.13 among others) will apply. Also don't forget paras 11.8 / 14.8 of the Insolvency PD which has important procedural points about making and serving the application.