Exhibit 5.1

Charles J. Bair

+1 858 550 6142

cbair@cooley.com

May 6, 2020

Kura Oncology, Inc.

3033 Science Park Road

Suite 220

San Diego, CA 92121

Ladies and Gentlemen:

We have acted as counsel to Kura Oncology, Inc., a Delaware corporation (the “Company”), in connection with the offering by the Company of up to 10,465,000 shares of the Company’s common stock, par value $0.0001 (the “Shares”), including up to 1,365,000 shares that may be sold pursuant to the exercise of an option to purchase additional shares, pursuant to a Registration Statement on Form S-3 (No. 333-232947) (the “Registration Statement”), filed with the Securities and Exchange Commission (the “Commission”) under the Securities Act of 1933, as amended (the “Act”), the prospectus included in the Registration Statement (the “Base Prospectus”), and the prospectus supplement relating to the Shares dated May 5, 2020, filed with the Commission pursuant to Rule 424(b) of the Rules and Regulations of the Act (together with the Base Prospectus, the “Prospectus.”)

In connection with this opinion, we have examined and relied upon the Registration Statement, the Prospectus, the Company’s Amended and Restated Certificate of Incorporation and Amended and Restated Bylaws, each as currently in effect, and originals or copies certified to our satisfaction of such records, documents, certificates, memoranda and other instruments as in our judgment are necessary or appropriate to enable us to render the opinion expressed below. In rendering this opinion, we have assumed the genuineness of all signatures; the authenticity of all documents submitted to us as originals; the conformity to originals of all documents submitted to us as copies; the accuracy, completeness and authenticity of certificates of public officials; and the due authorization, execution and delivery of all documents by all persons other than the Company where due authorization, execution and delivery are a prerequisite to the effectiveness thereof. As to certain factual matters, we have relied upon a certificate of an officer of the Company and have not independently verified such matters.

Our opinion herein is expressed solely with respect to the General Corporation Law of the State of Delaware. We express no opinion to the extent that any other laws are applicable to the subject matter hereof and express no opinion and provide no assurance as to compliance with any federal or state securities law, rule or regulation.

COOLEY LLP    4401 EASTGATE MALL    SAN DIEGO, CA    92121

T: (858) 550-6000 F: (858) 550-6420 COOLEY.COM

May 6, 2020 Page Two

On the basis of the foregoing, and in reliance thereon, we are of the opinion that the Shares, when sold in accordance with the Registration Statement and the Prospectus, will be validly issued, fully paid and nonassessable.

We consent to the reference to our firm under the caption “Legal Matters” in the Prospectus and to the filing of this opinion as an exhibit to a Current Report on Form 8-K to be filed with the Commission for incorporation by reference into the Registration Statement. This opinion is expressed as of the date hereof, and we disclaim any responsibility to advise you of any changes in the facts stated or assumed herein or of any changes in applicable law.

Sincerely,

COOLEY LLP    4401 EASTGATE MALL    SAN DIEGO, CA    92121

T: (858) 550-6000 F: (858) 550-6420 COOLEY.COM